<_)-k I' THEORY AND APPLICATIONS OF FINITE GROUPS BY G. A. MILLER rROFBSSOR OF MATHEMATICS IN THE UNIVERSITY OF ILLINOIS H. F. BLICHFELDT PROFESSOR OF MATHEMATICS IN STANFORD UNIVERSITY L. E. DICKSON PROFKSSOK OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO FIRST EDITION FIRST THOUSAND I NEW YORK ^ ^ > JOHN WILEY & SONS, Inc. \\y\/ V London: CHAPMAN & HALL, Limited ^y\{0 ^ 1916 (r'l Copyright, 1916^ BV G. A. MILLER, H. F. ULICHFKLDT and L. E. DICKSON m THK SCIENTIFIC FRESS KOMKItT ORUMMONO ANO COMPANY BROOKLYN. N. Y. To Catnille Jorban, whose fundamental investigations on the theory and .vpplications of finite groups enriched the subject to the extent of converting it into a fundamental branch of mathematics and furnished in a large measure the inspiration for the subsequent great activity in this field, this book is dedicated (by permission) PREFACE The aim of this book is to present in a unified manner the more fundamental aspects of finite groups and their applications, and at the same time to preserve the advantage which arises when each branch of an extensive subject is written by one who has long specialized in that branch. To secure unification, the three authors planned the book after various conferences and extensive correspondence, while each read and commented upon both the MS. and proof-sheets of the parts by the remaining authors. However, the influence of each author upon the other two has been mainly of editorial character, so that the individuality of the authorship of each part remains intact. Part I, written by G. A. Miller, gives in the first two chapters various concrete examples of groups and an elementary presen- tation of the most fundamental theorems on groups of sub- stitutions. These two chapters prepare the way, by easy stages, for the formal developments in the theory of abstract groups, to which the remaining six chapters of Part I are devoted. A reader who wishes to proceed as early as possible to the phases of group theory presented in Part II or Part III will find that the prerequisites for either of the latter parts are met by these first two chapters of Part I, with the exception that also 22, 27, 48 are needed for the last half of Part II, while 68 is needed for Part III. Chapter III is devoted to a development of fundamental theorems of abstract group theory and to the estabUshment of a logical connection between this theory and the theory of sub- stitution groups. A (1,1) correspondence between the abstract V vi PREFACE groups of finite order and the non-conjugate regular substitution groups is established in 27, where it is proved that every such abstract group can be represented as a regular substitution group and that any two simply isomorphic regular substitution groups must be conjugate. This (1, 1) correspondence is used frequently in the further development of the theory of abstract groups for the sake of furnishing concrete illustrations. The section in which it is established closes with a very simple recent proof of Sy low's theorem, and in 29 the interesting (^-sub- groups are considered for the first time in a textbook. The two most important categories of special abstract groups are doubtless the Abelian groups and the prime-power groups, and these two categories are treated in Chapters IV and V respectively. In the development of a theory of Abelian groups special emphasis has been placed on a determination of all their possible invariants, since these ' invariants seemed to offer the easiest means for studying various important questions in this theory. The Abelian groups which are groups of isomorphisms of cyclic groups receive especial attention in view of their appli- cations in number theory. Chapter V contains a considerable number of recent theorems. One of the most interesting of these is proved in 50 and establishes the fact that every poz- sible set of independent generators of any given prime-power group involves the same number of operators. Chapters VI and VII are more largely devoted to the developments due to the author than any other chapters of Part I. In the former of these two chapters various simple relations between two operators are considered and the cate- gories of groups which can be generated by two operators which satisfy these relations are determined, while the latter chapter is devoted to a study of groups of isomorphisms. The closing Chapter VIII of Part I deals with solvable groups and aims to be especially useful to those who seek a wide knowledge of the Galois theory of algebraic equations treated in Part III. Some of the exercises of Part I are due to questions asked by students and are intended to remove similar difficulties for the reader. In fact, this Part is based largely on a course of PREFACE va lectures given by its author at various times and the lecture notes were frequently changed so as to obviate difficulties which presented themselves to the students. Special thanks are due to Dr. E. A. Kircher for assisting the author in preparing these notes for publication, and to Professor W. A. Manning and Dr. Josephine E. Burns for valuable suggestions on the printer's proofs. Part II, written by H. F. Blichfeldt, seeks to give a more com- prehensive outline of the theory- of linear groups as developed up to the present moment than is contained in the pubUshed texts deaUng with this phase of group-theory. At the same time an attempt has been made to present this theory in as simple a manner as possible, consistent with brevity. Thus, in several places it has been deemed sufficient to indicate the method of proof of a general proposition by attending to a concrete case. From the outset the student is urged to work with the matrix form of a linear transformation ( 76). The practice thus gained is of great advantage throughout Part II; in particular, the more difficult sections of Chapter XIII will be mastered readily if the student has a clear mental image of the matrix form of the regular groups as depicted in 136 {M'). The introductory chapter (IX) and the chapter on binary groups (X) presuppose only the rudiments of ordinary group theory as given in 1-4, 6-9, 22, in addition to a few defi- nitions. By the aid of the Hermitian invariant ( 92), the the determination of the binary groups is here made to depend upon geometrical analysis, entirely with reference to Euclidean space. The following two chapters (XI, XII), exclusive of 116, 125, are based on articles published by the author, mainly in the Transactions of the American Mathematical Society , 1903-1911. Certain proofs have been recast and new theorems added. For the intelligent reading of these chapters and the next following, the principles of 1-22, 27, 48 should be well understood. The somewhat difficult theory of group-characteristics (Ch. XIII) has been developed along fairly easy lines, differing not only in arrangement, but also in methods of proof, from pre- yiii PREFACE uous expositions (cf. Dickson, Annals of Mathematics, 1902; and the references given on p. 257). The factorization of the group-determinant (the determinant of M, 136; cf. Weber, Algebra, edition 2, vol. II, pp. 207-218) follows as an obvious coroUar>' to 136 (i.e., Dm^Dw) and has therefore been omitted. It has been found impracticable to include a discussion of the arithmetical nature of the elements in the matrices ( 76) belonging to a finite group. The student may consult Burnside, Proceedings of the London Mathematical Society, ser. 2, vol. 4 (1907), p. 1 ; Schur, Mathematische Annalen, vol. 71 (1912), p. 355. Part III, written by L. E. Dickson, contains the essential principles of Galois' theory of algebraic equations (with emphasis on the condition for solvability by radicals), and extensive applications to geometrical questions. In the development of Galois' theory, the simpler case of numerical equations is treated before the case of equations whose coefficients involve variables, and only such rational func- tions are employed as are known to have denominators not equal to zero. In many of our discussions, the domain of the munbers re- garded as known undergoes successive enlargements; moreover, the initial domain is at our choice. Consequently there is an inherent ambiguity in the customary terms " cyclic equation," " simple equation," " abeUan equation," etc. An equation which is cyclic for one domain may not be cyclic for another domain. Such terms are therefore avoided in this text, being replaced by " equation whose group for the specified domain R is cyclic or simple, etc." For the sake of clearness, there is introduced the concept of solvability by radicals relatively to a domain R. The unqualified term " solvability by radicals " is reserved for the case in which the domain is that defined by the coefficients of the given equation. By the avoidance of the ambiguous terms mentioned and by the use of this generalization of solvability, we are able to , establish theorems the earlier published proofs of which were wholly inadequate. PREFACE ix The classic problems of the dupUcation of a cube, trisection of an angle, and the construction of a regular polygon of n sides by ruler and compasses are treated in a very simple manner by group theory. The problem of the determination of the nine points of inflexion of a plane cubic curs^e without singular points is treated adequately by group theory. The geometrical facts employed are not presupposed, but developed in an elementary manner. Similar remarks apply to the treatment of the problems of the determination of the 27 straight Unes on a general cubic surface and the 28 bitangents to a general plane quartic curve, and to the relation between these two problems. There is given an adequate basis for Hesse's and Cayley's notation for the 28 bitangents and an elementary derivation of the perfectly sym- metrical notation which arose from the theory of theta func- tions. An introduction is given to a recent advance in the appli- cations of groups to the question of the number of real roots of an algebraic equation or real elements of a geometrical con- figuration. Without finding the actual group G of the equation (usually a difficult task), it often suffices to examine the sub- stitutions of period 2 in a group having G as a subgroup. This is found to be sufficient for the case of the 27 Unes on a cubic surface, the 28 bitangents to a quartic curve, and for an ex- tensive class of problems on contacts of curves, so that the possible numbers of real elements are found with surprising ease. In order that the reader may secure early a thorough acquaint- ance with the concept of the group of an equation, there are given in the first two chapters of Part III seven sets of care- fully selected exercises, not too difficult for the beginner, in addition to the numerous examples treated in the text. A brief, but adequate, course in Galois' theory of equations is provided by 1-9, 12, 13, 17, 68, 140-171, which include 33 pages from Part I and 45 pages from Part III. TABLE OF CONTENTS PART I SUBSTITUTION AND ABSTRACT GROUPS CHAPTER I Examples of Groups and Fundamental Definitions SECTION PACB 1. The symmetric group of order six 1 2. The octic group 4 3. Generating substitutions of a group 7 4. The group of movements of plane figures 9 5. Congruence groups 10 6. Groups represented by matrices 13 Exercises 15 CHAPTER II Substitution Groups and Sylow's Theorem 7. Positive and negative substitutions 16 8. Commutative substitutions 18 9. Transforms of a substitution and of a substitution group 20 Exercises 23 10. Co-sets and double co-sets 24 11. Sylow's theorem 27 Exercises 30 12. Transitive groups and average number of letters in its substitutions 30 13. Intransitive substitution groups. . . .' 32 14. Substitutions which are commutative with each of the substitutions of a transitive group 35 Exercises 38 15. Primitive and imprimitive groups 38 Exercises 41 16. Groups involving no more than four letters ^ 41 17. Simplicity of the alternating group of degree n, nj^4 43 zi xii TABLE OF CONTENTS ncnoN , PACK 18. Gnjups of degree five 45 19. Holomorph of a regular group 46 Exercises 47 20. Class of a substitution group 47 Eierdses 50 CHAPTER III Fdndamental Definitions and Theorems of Abstract Groups 21. Introduction 51 22. Definition of an abstract group and a few properties of its elements 52 23. The cyclic group 54 Exercises 57 24. Properties of transforms 57 25. Construction of groups with invariant subgroups 59 26. The dihedral and the dicyclic groups 61 27. Representation of a group as a regular substitution group (>3 Cayley's theorem 64 Exercises 65 28. Invariant subgroups and quotient groups 66 29. Commutators, commutator subgroups and the 0-sub-groups 68 Exercises 73 30. Simply isomorphic groups 73 Exercises 76 31. Group of inner isomorphisms 76 32. Frobenius's theorem 77 Exercises 81 33. Representation of an abstract group as a transitive substitution group. . . 81 Exercises 84 34. Historical note 84 CHAPTER IV Abelian Groups 35. Invariants 87 36. Largest and smallest number of possible invariants 90 37. Number of elements of a given order 93 Exercises 94 38. Abelian groups of given orders 94 39. A special class of Abelian groups 95 Exerdaes 98 40. Sub-groups and quotient groups of any Abelian group 99 Exercises 101 41. Group of isomorphisms of an Abelian group 101 42. Groups of isomorphisms of the groups of order p^ 105 43. Abelian groups which are confomial with non-Abelian groups 107 Exercises 109 TABLE OF CONTENTS xiii SECTION PACS 14. Characteristic sub-groups of an Abelian group 109 Exercises 112 45. Non-Abelian groups in which every sub-group is Abelian 112 Exercises 114 46. Roots of the operators of an Abelian group 114 47. Hamilton groups 115 Exercises 117 CHAPTER V Groups Whose Orders are Powers of Prime Numbers 48. Introduction 118 49. Invariant Abelian sub-groups 120 Exercises , 122 50. Number of sub-groups in a group of order />"* 123 Exercises 127 51. Number of non-cyclic subgroups in a group of order p^, p>2 128 52. Number of non-cyclic subgroups in a group of order 2"* 129 Exercises 133 53. Some properties of the group of isomorphisms of a group of order p^ 134 Exercises 137 54. Maximum order of a Sylow subgroup in the group of isomorphisms of a group of order p** 137 55. Construction of all the possible groups of order p"* 138 Exercises 141 CHAPTER VI Groups Having Simple Abstract Definitions 56. Groups generated by two operators having a common square 143 Exercises 147 57. Groups of the regular polyhedrons 147 58. Group of the regular icosahedron 150 Exercises 151 59. Generalizations of the group of the regular tetrahedron 152 60. Generalizations of the octahedron group 154 Exercises ; 158 CHAPTER Vn Isomorphisms 61. Relative and intrinsic properties of the operators of a group 159 62. Group of isomorphisms as a substitution group 160 63. Groups of isomoq)hisms of non-Abclian groups 162 Exercises ''^ 04. Doubly transitive substitution groups of isomorphisms 104 .^3gjM xiv TABLE OF CONTENTS UCnON PACB 65. Groups of isomorphisms of the alternating and the symmetric groups.. . . 166 Exercises 168 66. Several useful theorems relating to groups of isomorphisms 168 67. Group of isomorphisms of a transitive substitution group 172 Exercises 173 CHAPTER Vni Solvable Groups 68. Introduction 174 Exercises 177 69. Series of composition 177 70. Groups involving no more than one non-cyclic Sylow subgroup 181 71. Groups whose nth group of inner isomorphisms is the identity 183 Exercises 184 72. Arbitrary choice of factors of composition 184 73. Groups of order f^(^ , P and q being prime numbers 185 74. Insolvable groups of low composite orders 186 Exercises ., 192 PART II FINITE GROUPS OF LINEAR HOMOGENEOUS TRANS- FORMATIONS CHAPTER IX Preliminary Theorems 75-82. Linear transformations; product of linear transformations 193 83-7. Groups of linear transformations 197 88. Change of variables 203 89. Characteristic equation 20.'> 90. Transitive and intransitive groups 206 91-3. Hcrmitian invariant; conjugate-imaginary groups; unitary form 207 94. Reducible and irreducible groups 210 95-6. Canonical form of a linear transformation and of abelian groups 212 CHAFFER X The Linear Groups in Two Variables 97-100. Introduction; geometrical analysis 215 101-3. The groups of the regular ixilyhedra 220 104-6. Invariants of the linear groups in two variables 224 TABLE OF CONTENTS rv CHAPTER XI Some Spectal Types of Groups section paob 106-7. Primitive and imprimitive groups 228 108-9. Sylow groups 230 110. On the group of similarity-transformations 233 CHAPTER XII The Linear Groups in Three Variables 111. Introduction 235 112-3. Intransitive and imprimitive groups 236 114-5. Groups having invariant intransitive or imprimitive subgroups 237 116. On roots of unity 239 117-23. Primitive simple groups 241 124. Primitive groups having invariant primitive subgroups 251 125. Invariants of the linear groups in three variables 253 126. Order of a primitive group in n variables; historical note 255 CHAPTER XIII Group Characteristics 127. Introduction 257 128-30. Number of invariants; conditions for transitivity 258 131-2. Equivalence 262 133-5. The sum of matrices; the product of characteristics 265 136. Number of independent elements in the group-matrix 268 137. Number of non-equivalent isomorphic groups 271 138. Groups of order p'^q are composite 272 139. Substitution groups of degree n and class n i are composite 274 PART III APPLICATIONS OF FINITE GROUPS CHAPTER XIV The Group of an Algebraic Equation for a Given Domain 140-1. Introduction, number domains 279 Exercises 280 142-4. Reducible and irreducible functions 280 Exercises 282 XVI TABLE OF CONTENTS SECTION PAGE 145. Functions with ! values. . 2<S2 14t>-7. Guloisiun resolvents . 283 148. The group 6' of an etjuation for a domain. . . 286 149. Characteristic proj>ertics of this proiip 6" . . 287 150. Transitive group 289 Exercises 291 151. Definitions of the roots for variable coefficients. . . 292 152. Function domains, equality . . 293 153. Group of the general equation-. . . 294 E.\ercises . . 29.5 154. Rational functions belonging to a group . . 296 155. Galois' generalization of Lagrange's theorem . . 2.)8 156. Effect t)n the group by an adjunction to the domain. . . 299 Exercises 300 CHAPTER XV StJFFiCTENT Condition that an Algebraic Equation be Solvable by Radicals l.')7. Solvability by radicals 301 158. Solution of a cubic equation 302 159. Resolvent equations and their groups 303 Exercises 305 1(50. Equations with a regular cyclic group 306 161-3. Cyclotomic equations, Gauss' lemma 308 164. Sufficient condition for solvability by radicals 310 165. Solution of a quartic equation 312 Exercises 314 CHAPTER XVI Necessary Condition that an Algebraic Equation be solvable by Radicals 166. Galois' criterion for solvability -5 1 ' 167. Theorems of Galois, Jordan, H6lder 317 CHAPTER XVn Constructions with Ruler and Compasses 168. Some celebrated problems of Greek origin. 169. Analytic criterion for constructibility. 170. Triscction of an angle 171. Duplication of a cube 172. Regular polygons 321 321 323 323 323 TABLE OF CONTENTS xvii CHAPTER XVIII The Inflexion Points of a Plane Cubic Curve section pack 173. Homogeneous coordinates, Euler's theorem, singular points 327 174. Hessian curve 329 175. Inflexion points, inflexion triangles 330 176-8. Group of the equation for the inflexion points 33:j 179. Real points of inflexion 341 CHAPTER XIX The 27 Straight Lines on a General Cubic Surface and the 28 bitangents to a general quartic curve 180. Existence of the 27 lines on a cubic surface 343 181. Double-six configuration 345 182. The 45 triangles on a cubic surface 346 183. Group of the equation for the 27 lines 347 184. Relation between cubic surfaces and quartic curves .'J51 Exercises 353 185. Steiner sets of bitangents to a quartic curve 354 186. Notation of Hesse and Cayley for the bitangents 357 187. Group containing the group for the 28 bitangents 362 188. Number of real bitangents to -a. quartic curve 365 189. Number of real lines on a cubic surface 366 190. Actual determination of the group for the bitangents 367 191. Symmetrical notation for the bitangents 372 192. Further problems of contacts of curves 375 CHAPTER XX MoNODROMiE Group 193. Monodromie group M of F(z, k)=0 378 194. M an invariant subgroup of the Galois group of F 378 195. Applications of monodromie, differential equations 379 196. Quintic equations, form problem 381 PART I* SUBSTITUTION AND ABSTRACT GROUPS CHAPTER I EXAMPLES OF GROUPS AND FUNDAMENTAL DEFINITIONS 1. The Symmetric Group of Order Six. There are six move- ments of a plane which transform into itself a given equilat- eral triangle situated in this plane. These are the rotations about the center of the triangle through angles 0, 120, 240, and the rotations through 180 of the plane about an altitude of the triangle. If we represent the vertices of this triangle by the letters a, b, c, the results of these six movements, which include the identity, are represented by the following figures: abb c c a b a a c Figs. 1-6. All of these figures may be obtained from any one of them by interchanging the letters in every possible manner. Such Interchanges of letters are called substitutions on these letters. Various symbols have been employed to represent substi- tutions. According to one of the oldest and most elementary types of symbols, the given six substitutions are represented as follows: /abc\ /abc\ /abc\ /abc\ /abc\ /abc\ \abc/^ \bca/' \cab/' \bac/' \acb/' \cba/' This part was written by (J. A. Miller. 2 EXAMPLES OF GUOUPS; DEFINITIONS (Ch. I This notation implies that each letter is to be replaced by the one just below it in the same symbol. The same substitutions, in order, are commonly represented by the following briefer symbols: * 1, ahc, ach, ah, he, ac. This notation implies that each letter is to be replaced by the one which follows it in the same symbol, the last being replaced by the first. Letters which are not replaced are omitted in this notation, and the symbol for unity is used to represent the identity; that is, the substitution in which every letter is replaced by itself. It is easy to verify the fact that any two of these substitu- tions, when performed successively, are equivalent to a single one of them. For instance, if we first apply ah and then ac the result is the same as if we had applied ahc only once. The process of combining (composing) two substitutions into one is called multiplication, and it is denoted by the common symbols for multiplication. Hence ahc is said to be the product of ab and ac. Since ac-ah = acb, and ab-ac = abc, it results that the commutative law of multiplication is not always satisfied as regards the multiphcation of substitutions. A set of distinct substitutions, which has the property that no additional substitution can be obtained by multiplying successively each substitution of the set into all the substitu- tions of the set, is called a substitution groups Hence the given set of six substitutions constitutes a substitution group. The number of the distinct substitutions of a group is called the order of the group and the number of the distinct letters in its substitutions is the degree of the group. The totality of the possible ! substitutions on n letters evidently constitutes a * These symljols have been called the normal forms of substitutions, J. de S^uicr, Croupes de Substitutions. 1912, p. 3. They are often inclosed in parentheses. t The term group in this technical sense is due to E. Galois (1811-32). The statement that the term group was not used before 1870 with its present technical meaning, which is found in the Encydopadia Britannica, eleventh edition, vol. 22, p. 620, is incorrect. S ll SYMMETRIC GROUP OF ORDER SIX 3 substitution group, and this is known as the symmetric group of degree . The symmetric group of degree n exists for every value of , > 1, and it includes every possible group on these letters. There may, however, be other groups on n letters. In fact- it is easy to verify that the three substitutions 1, abc, acb constitute a second group on the three given letters. Hence we have found two substitution groups on three letters, and it can be verified that no other groups involving these three letters are possible. That is, if a set of substitutions involving the three letters a, b, c constitutes a group, this group is of order 3 or of order 6, and there is only one such group of each of these orders if we regard two substitution groups identical when they differ only as regards the letters involved. This group of order 3 could have been found by trial combinations of the six possible substitutions on three letters, but it results also directly from the fact that it corresponds to the rota- tions of the given triangle about its center through the angles 0, 120 and 240". As this group of order 3 is contained in the symmetric group of order 6 it is called a subgroup or a divisor of the latter. The identity is a subgroup of every group. The only other sub- groups of this symmetric group are of order 2. There are three such subgroups, viz., 1, ab; 1, ac; 1, be. Hence the symmetric group of order 6 has four subgroups besides the identity. The last three of these correspond to the groups of movements through the angle v around the lines of sym- metry of the given triangle. The symmetric group of order 6 could also have been found by considering the possible permutations of three variables which do not alter a symmetric function of these variables. A simple instance of such a function is the following: x-\-y+z. 4 EXAMPLES OF GROUPS; DEFINITIONS (Ch. I The given group of onler '6 may be obtained from a considera- tion of the permutations which leave formally unaltered the function (x-y){y-z)(z-x) of the independent variables x, y, z. 2. The Octic Group.* There are eight movements of a plane which transform into itself a square situated in this plane. If we represent the vertices of this square by the letters a, b, c, d, the results of these eight movements are represented by the following figures: d c ha ad c h a b c d be do, d a b c c d a b c b a d b Figs. 7-14. a d c The corresponding eight substitutions are as follows: 1, ac-bd, abed, adeb, ae, bd, ab-cd, ad- be. It should be observed that three of these substitutions are separately the products of two substitutions on distinct sets of letters. For instance, the substitution ae-bd "mplies that a and c are replaced by each other, and that b and d are also replaced by each other. The symmetric group on four letters is composed of 24 substitutions. Hence the octic group is composed of one- third of the possible substitutions on four letters. It con- tains three subgroups of order 4, viz., 1, ac-bd, abed, adeb; 1, ac-bd, ac, bd] 1, ac-bd, ab-ed, ad-be. The first of these corresponds to the rotations about the center of the square through the angles 0, 7r/2, x, and 37r/2. The This group is also known as the group of the square, and as the dihedral group of order 8. It is given in 108 of the noted article by J. L. Lagrange entitled, " Reflexions sur la resolution algebrique des Equations,' which was published in the Nouveaux Mimoires de VAcadlmk royale de Berlin, 1771. 21 OCTIC GROUP 6 second corresponds to the movements through the angle v around the diagonals, and their combinations. The third cor- responds to the movements through the angle x around the lines joining the middle points of opposite sides, and their combinations. The three subgroups of order 4 have the subgroup 1, acbd in common. There are four other subgroups of order 2 in the octic group, viz., 1, ac; 1, bd; 1, ab-cd; 1, adbc. Hence the octic group has three subgroups of order 4 and five subgroups of order 2 besides the identity. It can be verified that no other subgroup exists in this octic group. Another illustration of this group may be based on the function of four variables which is transformed into itself by the following eight sub- stitutions 1, X\Xz'X2PC4^, XiX2X:iX4^, ^^1X4^^3X2, X\Xz, X2X4, X\X2'X3Xa, X\Xa' X2X:i. These substitutions constitute the octic group, since they can be changed or transformed into those of the given octic group by the substitution x\a-X2b-xzC-XAd. Substitutions and substitution groups which can be trans- formed into each other by an interchange of letters are said to be conjugate. The remaim'ng sixteen substitutions on these four variables transform the function xiXi-{-X2X\ into one of the following two functions : X\X2-\rXzXA, XiXJk-\-X2XZ' Each of these two functions belongs to the group of all the substitutions on these four letters which transform the func- tion into itself. These two groups are distinct from each other and also from the octic group obtained above. In fact, 6 EXAMPLES OF GROUPS; DEFINITIONS (C'h. I the three groups are' conjugate,* and they have in common the following subgroup of order four: 1, X\X2'XzXa, X\Xz'X2XAy X\Xa' X2Xi. Hence the symmetric group of degree four contains three con- jugate octic groups. These octic groups have four substitutions in common, and therefore they involve sixteen distinct substi- tutions. An interesting illustration of the applications of the octic group in elementary trigonometry is furnished by the opera- tions of finding the complement and the supplement of an angle a, and of the angles obtained from a by these operations. We thus obtain the following eight geometric angles and no more: a, 90 -a, 180 -a, 90 +a, 270 -fa, -a, 270 -a, 180 +a. 270-a 270+a Fig. \T>. * Two ronjugalc substitution groups are often regarded as the same group, but when they arc subgroups of a given group they arc often said to be distinct whenever one contains at least one substitution which is not in the other. In fisting the possible substitution groups on a given number of letters conjugate groups arc regarded as identical. 3] GENERATING SUBSTITUTIONS 7 Each of these eight angles may be regarded as representing also the operation by means of which it can be derived from a. From this point of view these angles constitute a group of order 8. That this is actually the octic group may be seen as follows: If the angle a is represented by a point on a unit circle, the process of finding the complement of a is equivalent to reflecting a on the bisector of the first and third quadrants, since the geometric meaning of subtraction is a reflection on the point midway between and the minuend. Hence the process of finding the supplement of any angle a is equivalent to reflecting on the Y-axis the point representing a. Since these two kinds of reflections transform into itself the square whose sides are parallel to the coordinate axes, inscribed in the unit circle, it results that the group represented by the eight angles which may be obtained from a by the operations of finding the complement and the supplement is the octic group. One important difi^erence between the group of movements of the triangle and the group of movements of the square should be emphasized here. In the former case we obtain exactly the same substitution group on the letters a, b, c no matter how these letters are arranged so as to represent the vertices. In the latter case, some possible rearrangements give rise to a differ- ent substitution group. In fact, by starting with different arrangements we may obtain three possible conjugate octic groups on the letters a, h, c, d by movements which transform the figure as a whole into itself. We thus obtain another ele- mentary illustration of the important concept of conjugate substitution groups, or of subgroups conjugate under the symmetric group. 3. Generating Substitutions of a Group. The different distinct powers of any substitution constitute a group, which is called cyclic. If it is not possible to find at least one substi- tution in a group such that all the others are powers of it, the group is non-cyclic. The s^-mmetric group of order 6 and the given conjugate octic groups are non-cyclic. Each of the latter groups involves one cyclic and two non-cyclic subgroups of order 4. 8 EXAMPLES OF GROUPS; DEFINITIONS [Ch. I A substitution whose powers give all the substitutions of a group is said to generate this cyclic group, and the order of this cyclic group is equal to the order of the substitution. It is always possible to select such a generating substitution in more than one way except when the order of the cyclic group is 1 or 2. For instance, the cyclic group of order 4 1, ac-hd, abed, adcb is generated by abed as well as by adcb, and the cyclic group of order 3 1, abc, aeb has also two generating substitutions. In general, a substitution group is said to be generated by a set of substitutions provided all of the substitutions of the group can be obtained by combining those of the set. The least number of substitutions that can generate a non-cychc group is two. Each of the non-cyclic groups which have been con- sidered thus far can be generated by two of its substitutions. For instance, the symmetric group of order 6 can be generated by any one of its three possible pairs of two distinct substi- tutions of order 2. It can also be generated by any one of the six distinct pairs composed of one substitution of order 2 and one of order 3. Hence the symmetric group of order 6 has nine distinct pairs of generating substitutions. The octic group cannot be generated by every possible pair of distinct substitutions of order 2, since some such pairs gen- erate only four substitutions. In fact, it is easy to verify that only four out of these ten possible pairs generate this group, while each of the remaining six generate a group of order 4. The square of the substitutions of order 4 cannot be used as one of a pair of generating substitutions of the octic group, but every other substitution besides the identity of this group occurs in such a pair. Hence it is not difficult to verify that there are exactly 12 possible pairs of generating substitu- tions of the octic group. Any set of substitutions on n letters generates some sub- stitution group on these letters, which is contained in the sym- 4) GROUPS OF PLANE FIGURES 9 metric group of degree n. It is not necessary that each of these substitutions should actually involve all of these n letters. For instance, it may be found by trial that the two sub- stitutions abc and abd generate a group of order 12, while the two substitutions abc and ad generate the symmetric group of order 24. There is no upper limit for the order of a group which can be generated by two substitutions if these substitutions be chosen arbitrarily and their degrees are not limited. A set of X substitutions Si, S2, . . . , Sx oi a. finite substi- tution group G is called a set of generators of G provided there is no subgroup in G which includes each of these substitutions. When these substitutions satisfy the additional condition that G can be generated by no X 1 of them, the set is said to be a set of independent generators of G. Such a set can usually be chosen in many different ways. 4. The Groups of Movements of Plane Figures. The sym- metric group of order 6 and the octic group are special cases of the groups of movements of regular polygons. The regular polygons of n sides are evidently transformed into themselves by the cyclic group of order n which is generated by the sub- stitution corresponding to the permutation of the vertices when the polygon is rotated around the center through the angle 2ir/. They are also transformed into themselves by n substitutions of order 2 which correspond to the permutation of the vertices when the polygons are rotated successively through the angle x around their different lines of sjTnmetry. As no other movements transform these polygons into them- selves, it results that the group of movements of a regular polygon of n sides is of order 2. According to a common definition of regular polygons there is only one regular polygon of 3, 4, or 6 sides, but there are two regular polygons of five sides, as may be seen by connecting alternate vertices, and there are three such poly- gons of 7 sides. In fact, it is not difficult to see that the num- ber of such regular polygons of n sides is equal to one-half the number of generating substitutions of the cyclic group of order . All of these regular polygons of n sides belong to the same 10 EXAMPLES OF GROUPS; DEFINITIONS (Ch. 1 substitution group, but this group can be most easily studied by means of the polygon obtained by connecting successive points dividing the unit circle into n equal parts. It contains n or + l substitutions of order 2 according as n is odd or even. If we consider the general problem of transforming a system of n, n>2, non-collinear points in the rigid plane among them- selves, it is important to observe that all such transformations leave a given point invariant, viz., the center of the smallest circle that circumscribes the system of points in question. Hence the possible movements are restricted to rotations around this center, or rotations through the angle tt on a line passing through this invariant point. The former rotations must constitute either exactly half or all of the possible movements. In the case of the general parallelogram they evidently con- stitute all of the possible movements, while they constitute exactly half of the possible four movements in the case of the general rectangle. This result may also be expressed as follows: A plane figure either has no line of symmetry or it has as many lines of symmetry as it admits different plane rotations around the center of its smallest circumscribing circle. In the former case its group of movements is cyclic, while it is non-cyclic in the latter case. 5. Congruence Groups. It will be proved later that every finite group can be represented as a substitution group. Many groups present themselves naturally in different forms and hence it is desirable to study groups represented in different ways. For the present we shall, however, study these groups by means of substitution groups. Suppose that the first m l p)Ositive integers, together with zero, are combined by addition, and the sums are replaced by their least positive or zero residues modulo tn. It is clear that no new numbers are obtained in this way. If any number is added separately to itself and to each of the others, the tn numbers will be permuted according to a substitution which may be associated with this added number. The substitutions which are thus associated with all these numbers constitute the cyclic group of order m. 5] CONGRUENCE GROUPS . 11 The substitution which is associated with is the identity, while the one which is associated with 1 must be a generating substitution of the cyclic group of order m. If k is any one of these m numbers, the substitution which corresponds to k will be of order m/d, where d is the highest common factor of m and k. Two substitutions whose product is the identity are said to be the inverses of each other. Hence the inverse of the substitution which corresponds to k is the one which corresponds to m k. Moreover, if the (f>{m)* positive integers which are prime to m and not greater than m are combined by multiplication, and the products are reduced mod ulo m, no additional numbers are obtained by this operation. For instance, if w = 8 and we multiply the four numbers -^ 1, d, 5, 7 in succession by 1, 3, 5, 7, we obtain the following non-cyclic substitution group of order 4: ^rr. urv v;>()W 1, 13-57, 15-37, 17-35 If OT = 5 and we combine the numbers 1, 2, 3, 4, by multiplica- tion, there results the following cych'c group of order 4: 1, 1243, 1342, 14-23 The two given types of groups furnish illustrations of the following very important category of congruence groups, where p is a prime number, viz., the groups formed by all the f)ossible linear substitutions of the following form: X =ax-\-b (mod PH , ^ . ^ ^ , 6 = 0, 1, 2, ... , p-l. * This symbol was first used by Gauss to represent the present concept. It is called the lolietU of m according to Sylvester. The French called it the indicator of m, and the Germans commonly call it the FmIct ^-function of m. t This group is sometimes represented by the following general substitution: /O I 2 ... ^ - 1 \ \b a+b 2a+b . . . (Jt-l)a+b/ 12 EXAMPLES OF GROUPS; DEFINITIONS [Ch. I It is easy to find the oMer of such a substitution as follows: Let 5 represent the substitution x'=ax-\-b. Then s^ is x' = a^x-\-ia-\-l)b, and s'isx'=a'x+(a'-i+a'-2+ . . . +a + l)6. If s' = l, it is necessary that a'"=l (mod p). This is equiv- alent to a'-l = (a-l)(a'-^+a'-^+ . . . +a + l)=0 (mod />). When a^l (mod p), it results that a necessary and suf- ficient condition that 5'" = 1 is that a^ = l (mod p). That is, the order of s is the exponent to which a belongs (mod p) except when a = l (mod p). In the latter special case, j is clearly of order p when b is not zero. It should be observed that the order of 5 is independent of the value of b except when a = \ (mod p). Hence the con- gruence group under consideration has p1 substitutions of order p, and p6{d) substitutions of order d, d being any divisor of p l, including /> 1 but excluding 1. The order of this group is therefore p{p l), as is also evident from its definition. It is commonly known as the mclacyclic group of order p{p \). The term metacyclic has also been used by H. Weber in his Lehrbuch der Algebra, 1895, page 598, to define a more general category of groups, but the definition given here is the older one and is still commonly used. A particular type of linear groups considered by E. Galois consists of the w* operations which may be represented by I 2i, 22, . . . ,zt zi-f-ci, 22+C2, ., Zt-\-Ct \ (mod m) where each 2< is replaced by Zt-{-Ci (mod m). In a more explicit notation, we have z'i=Zt-\-Ci (mod m), i=l, 2, . . . , k. This group of order w* was called by A. L. Cauchy the group of arUhmetic substiiulions. In particular, when m = k = 2 we S6] GROUPS REPRESENTED BY MATRICES 18 obtain the non-cyclic group of order 4 noted above. This general group is clearly generated by the k substitutions of which the ith (for i = 2, 3, . . . , ife-1) is z'i=zi, . . .,z\-i=Zi-i,z't = Zi-\-l,z\+i=Zt+i, . . .,2't=Z4(modm). For i=l and i = k this substitution becomes respectively z'l=Sl + l, 2^2=22, . . ., 2't = Zt z'i = 2i, . . ., z'_i = 2t_i,2't = 2t4-l (mod m). and Each of these k substitutions generates a cyclic group of order m, and the entire group is said to be the direct product of these k independent cycHc groups. 6. Groups Represented by Matrices. If two matrices of order n are multiplied together in the ordinary manner * there results a matrix of order n. It may happen that the elements of the matrices are of such a nature that only a fim'te number of different matrices result when a given set of them are com- bined by multiplication in every possible order. For instance, the six matrices 1 -1 1 -1 1 -1 1 -1 1 } -1 > 1 -1 > 1 J -1 1 > -1 and the six substitutions 1, abc, acb, ah, ac, be may be put into a (1, 1) correspondence in the given order so that hke products always correspond. That is, the given matrices combined under multiplication constitute the sym- metric group of order six, or the group of an equilateral triangle. We thus meet this group in another very important and exten- sive field of mathematics, viz., the theory of matrices. It may be observed that the. first three of these matrices con- stitute the subgroup of order three and they can therefore be put into a (1, 1) correspondence, as regards multiplication, with the cube roots of unity. * For a definition of the product of two matrices cf. BAcher, Inirodudum t9 Higher Algebra, 1907, p. 62. 14 EXAMPLES OF GROUPS; DEFINITIONS [Oh. 1 It is not difficult to verify that the following six matrices also constitute a group of order six as regards multiplication: 1 1 > -1 1 -1 > -1 1 -1 > -1 0| 0-1 ' 1 -1 1 > 1 -1 1 This group of order six can, however, not be put in a (1, 1) corresp>ondence with the one considered in the preceding para- graph, although the first three matrices are the same in the two groups. The present group of order six is generated by a single matrix and hence it is cyclic. As a generating matrix we may use either one of the last two matrices. It may also be observed that multiplication is commutative in the present group, while this is not the case in the group of the preceding paragraph. The matrices of the present group can be placed in a (1, 1) correspondence with the six sixth roots of unity as regards the operation multiplication. The four matrices 1 1 ) -1 0-1 > 1 0-1 > -1 1 evidently serve as another illustration of the non-cyclic group of order 4 when they are combined by multiplication, while the four matrices 1 1 > -1 0-1 > 1 -1 > -1 1 combined according to the same operation, constitute the cyclic group of order 4. As an instance of matrices which constitute a cyclic group of an arbitrary order we may observe that the matrices 1 1 1 1 1 1 2 1 1 3 1 1 m-1 1 constitute the cyclic group of order m when they are combined as to multiplication, and the elements are reduced modulo m. 61 GROUPS REPRESENTED BY MATRICES 16 EXERCISES 1. In a letter to C. Hennite [Paris Comptes Rendus, vol. 49 (1859), p. 115], E. Betti states that the following substitutions: z'=4z, 2'=-, 2'=3^ (mods) t s 1 generate a group of order 12. Verify this statement, and prove that the first two of these substitutions generate the non-cyclic group of order 4. Either of the first two of the three given generating substitutions could be omitted without affecting the resulting group. 2. Prove that the transformations of the form z' = (mod 3), o, yz+i /3, 7, 5 being integers such that aS ^=1 (mod 3), constitute a group of order 12 which has the same number of elements of each order as the one in Example 1. Cf. F. Klein, Mathematische Annalen, vol. 14 (1879), p. 418. 3. By subtracting a given number n from unity and dividing unity by n, and then performing the same operations successively on the result- ing numbers, it is possible to obtain, in general, six different numbers. The only exceptions occur when n has one of the following 8 values: 1, 2, ; 1,0, oo; ^(1V 3). If n represents an anharmonic ratio of four points, then the six values of n obtained by the given operations represent all the values of this ratio. CHAPTER n SUBSTITUTION GROUPS AND SYLOW'S THEOREM * 7. Positive and Negative Substitutions. The symbol aia2 ... On is called a cyclic suhslitulion of degree n, and it implies that each of these letters is replaced by the one which follows it, and that is replaced by ai. That is, it implies a cyclic interchange of its letters, and any one of these letters may be written first without changing the rrieaning of the substitu- tion. Any permutation of a set of m distinct letters can evi- dently be effected by one or more cyclic substitutions on these letters. If a substitution consists of more than one cycle it is said to be non-cyclic and its various cycles are usually sepa- rated by periods, and the number of letters in all these cycles is the degree of the substitution. For instance, a\a2az-aAas denotes a substitution of degree 5 which implies a cyclic inter- change of the letters ai, 02, as in order, and also an interchange of the letters a^, as- A cyclic substitution which involves only two letters is caUed a transposition. The fact that every possible substi- tution is a product of transpositions results directly from each of the following equations: aia2a3 . . an = aia2'aia3' . . . -aian aiChi-Onan-i- . . . a3a2 A given substitution can be factored into transpositions in an infinite nunr^r of different ways, but the number of the transpositions intQ which a particular substitution can be C i The general theory of substitutions and substitution groups is developed from the beginning in the present chapter. A few definitions given in the pre- ceding chapter are, however, not repeated. 16 7] POSITIVE AND NEGATIVE SUBSTITUTIONS 17 factored is either always even or it is always odd. We pro- ceed to prove this important elementary theorem. Let 5 represent any substitution on m letters and suppose that ^ has been factored into transpositions in various ways. Each of these transpositions changes the sign of the following determinant: * A = 1 ai ai^ . . . ai*^~^ which is not identically zero. As the various sets of tran- positions which are equivalent to s must have the same effect on A as 5 has, it results that the number of transpositions in every set is odd if 5 transforms A into A, and this number is even if 5 transforms A into itself. A substitution is said to be positive if it can be factored into an even number of transpositions. If it can be factored into an odd number of transpositions it is called negative. For instance, abc = ab-ac is positive, and abed = ab ac ad is nega- tive. The product of two positive substitutions is positive and the product of two negative substitutions is also positive. Hence a product of a set of substitutions is positive or negative according as it involves an even or an odd number of negative substitutions. If a substitution group involves a negative substitution then exactly one-half ^of its substitutions are negative. In fact, if all its positive substitutions are multiplied into this negative substitution, all of these products are distinct and negative. Hence it has at least as many negative substitutions as positive ones. On the other hand, if this negative substi- tution is multiplied into all of its negative substitutions, all these products are distinct and positive. Hence it has at least as many positive substitutions as negative ones. In other * This determinant is known as the determinant of Vandermonde or o( Cauchy. It is equal to the product Uioi-qt); i, k=l, 2, . . . , m; i>k. 18 SUBSTITUTION GROUPS [Ch. II words, either all the substitutions oj a substitution group are posi- tive or exactly half oj tliem are positive. W^henever a group contains negative substitutions it con- tains a subgroup of half its own order, composed of its positive substitutions. In particular, the symmetric group of degree n contains a subgroup of order !/2 which is composed of its positive substitutions. This subgroup is called the alternating group of degree n. Hence there are at least three distinct groups of degree n whenever w>3, viz., the group generated by a cyclic substitution on n letters, the alternating group, and the symmetric group. It will be proved that other groups of degree n exist for every value of w>3. The cyclic substi- tution of degree^ is positive or negative according as n is odd or even. The product of two transpositions which have a common letter is always of the form abc. A positive cyclic substitu- tion is always the product of substitutions of the form abc, since it is the product of an even number of transpositions having a common letter. A substitution composed of two negative cyclic substitutions is also the product of substitu- tions of the form abc. In fact, such a substitution may be regarded as the product of two distinct sets of transpositions such that all the transpositions of each set have a common letter and such that each set involves an odd number of trans- p)ositions. Hence it remains only to observe that a substitu- tion composed of two transpositions having no letter in common is the product of substitutions of the form abc. This fact results directly from the pro'duct ab-cd = acb bdc. That is, every possible positive substitution is the product of ^sub- stitutions of the form abc. 8. Commutative Substitutions. Let s and / represent two substitutions. If st = ts, these two substitutions are said to be commutative. For instance, il s = ab-cd and t = acbd, it is easy to verify that st = ts = adbc. On the other hand, if s = ab-cd and t\=bc, it results that st\=acdb while tis = abdc. Hence 8] COMMUTATIVE SUBSTITUTIONS 19 the two substitutions ab-cd, acbd are commutative, but ab-cd is not commutative with be. Two substitutions which have no letter in common are always commutative. It is often necessary to find all the substitutions on certain letters which are commutative- with a given substitution. The solution of this problem is based on finding all the substitutions on the letters ai, a2, . . . , an which are commutative with the cyclic substitution Si=aia2 ... On- It is clear that Si is commutative with all of its powers, and hence Si is commu- tative with at least n substitutions, including the identity, on the letters ai, a2, . . . , On- All substitutions which are commutative with si must also be commutative with Si"\ where ai is any positive integer. Suppose that /2 is a substitution on the letters ai, a2, . . . , 0% which is commutative with si but is not a power of Si. It is evident that /2 must involve each of the letters ai, a2, . . . , a. Hence we may suppose that h = CLiaa , where a is one of the numbers 2, 3, . . . , w. Since 5i"~^=aiaa . . . , it results that t2Si*'^^~'' is a substitution which is commutative with 5i, does not involve ai, and is not the identity. That is, we arrive at an absurdity by assuming that more than n sub- stitutions on the letters ai, a2, . . . , On are commutative with si. This proves the theorem: The only substitutions on n letters which are commutative with a cyclic substitution on these letters are the powers of this cyclic substitution. If a substitution 52 is composed of X cycles such that no two of these cycles involve the same number of letters, then all the substitutions on the letters of S2, which are commutative with S2 must also be commutative with each cycle of s-y. The number of the substitutions which are commutative with 50, and involve only letters contained in 52, is therefore equal to the product of the orders of the cycles of 52. For instance, the substitutions which are commutative with the following substitution abcde-fgh, and involve only its eight letters, constitute a substitution group of order 15. 20 SUBSTITUTION GROUPS [Ch. II A substitution 53 which involves k equal cycles is commuta- tive with substitutions which permute these cycles according to the symmetric group of degree k. If each of these cycle- involves n letters, the substitutions on these kn letters which are commutative with S3 must therefore constitute a substi- tution group of order n^-kl. For instance, the substitutions on the nine letters involved in the following substitution abc'def-ghi, and which are commutative with this substitution, constitute a group of order 27-6 = 162. 9. Transforms of a Substitution and of a Substitution Group. If s and / represent any two substitutions, it is possible to find a third substitution by means of the operation s~Hs = ti, where s~^ represents the inverse of s. The substitution /i is called the transform * of / as regards s. There is a very simple rule for deriving /i if s and / are given. We proceed to develop this rule. Suppose that t= . . . aap ... It is not assumed that a^ or a^ actually appears in s when s is written in the normal form, since a'^ may be identically equal to da and a'fi may be identically equal to a^. Hence the given notation is entirely general. It is easy to see that t\= . . . dad p . . That is, to obtain the transform of t as regards s we simply replace each letter in t by the otie by which s replaces this letter. f For instance, if seabed- ef and t = ab-ce, then s~Hs = bc-df. In This transformation is fundamental in the theory of groups. Many of its properties were developed by E. Betti in 1852; Annali di Scienze malematichc efisiche, vol. 3, p. .'.'). t This simple method to find the transform of a substitution is found in C. Jordan's thesis, Paris, 1860, p. 14. 9] TRANSFORMS OF SUBSTITUTIONS 21 particular, if a substitution contains a certain number of cycles of a given order, each of its transforms contains the same number of cycles of this order. A necessary and sufficient condition that t = ti is that s and t are commutative, since the equation s~Hs = t implies that ts = st and conversely. Each of two commutative substitutions is transformed into itself by the other. If a substitution is transformed into itself by all the substitutions of a group it is said to be invariant under this group. When this substitu- tion belongs to the group it is said to be an invariant substitu- tion of the group. The identity is invariant under every group and is an invariant substitution of every group. As an instance of another invariant substitution it may be observed that ac-bd is invariant under the following octic group : 1, ab-cd, aC'bd, ad- be, ac, bd, abed, adeb. The symmetric group of order 6 contains no invariant substi- tution besides the identity. If each substitution of a group G is transformed by means of a substitution s there results a group G\ which is called the transform of G with respect to s, or the conjugate of G with respect to s, and is represented by the symbol s~^Gs = Gi. Conjugate substitutions and conjugate groups are also called similar. When Gi=G the substitution 5 is said to transform G into itself, and G is said to be invariant under s. Every group is invariant under its own substitutions. A subgroup which is invariant under all the substitutions of a group is called an invariant, or self -conjugate, subgroup. If a group involves negative substitutions all of its positive substitutions consti- tute an invariant subgroup. In particular, the alternating group of degree n is an invariant subgroup of the symmetric group of this degree. Suppose that s is transformed into itself by some but not by all of the substitutions of a group G. The substitutions 22 SUBSTITUTION GROUPS [Ch. II of G which transform s into itself form a subgroup // of G. If / is any substitution of G which is not in H then all the sub- stitutions obtained by multiplying / on the left by a substi- tution of // will transform s into the same substitution. For, if /i is any substitution of H, we have the equation Moreover, if t'-^st' = r^st, it results that s = t'r^sU'~K That is, U'~^ is in H. Hence its inverse t't~^ is also in H. That is, /' is one of the substitutions obtained by multiplying t on the left by some substitution of H. From what precedes it results that all the substitutions of G can be divided into equal sets such that each set is composed of all the substitutions of G which transform s into the same substitution. The totality of the substitutions into which s is transformed under G forms a complete set of conjugates of s under G. The number of different substitutions in a com- plete set of conjugates under a group is a divisor of the order of the group. A complete set of conjugate subgroups under G is defined in a similar manner. Incidentally we proved above that the order of G is divisible by the order of its subgroup H. We proceed to prove that the order g of G is divisible by the order k of any subgroup K* In fact, all the substitutions of G can be written in the form of a rectangle, whose first line is composed of the substitutions of K, as follows: 1, 52, ^3, . ' ., St /2, ^2^2, Szt2, . . ., Stt2 t\, S2t\, 53/x, . . ., Stt\ In this rectangular array the substitutions t2, . . . , i\ are any substitutions of G which do not occur in any of the preceding rows. * If a group IS represented by a capital letter the corresponding small letter usually represents the order of the group. 9] TRANSFORMS OF SUBSTITUTIONS 23 To prove that ^ is a divisor of g it is only necessary to prove that no substitution of G can occur twice in such an array. This fact can be easily proved as follows : If sjy = Sfity {a, ^%k), th en 5 = 5^ ; and if sJy = Spti{b>'y)j then Sfi~^sJy=Sa'ty = ti{a%k). Hence it results that no substitution of G can occur twice in such an array and we have established a fundamental theorem, known as the theorem of Lagrange, which may be stated as follows : The order of a group is divisible by the order of each one of its subgroups. This may be regarded as the most important theorem of group theory. It will appear that a subgroup has properties similar to those of a modulus in number theory. Hence a sub- group is sometimes called a modulus of the group. The quotient obtained by dividing the order of a group by the order of a subgroup is called the index of this subgroup under the group. The index of a subgroup is therefore always a positive integer. EXERCISES 1. The order of a group is divisible by the order of each of its sub- stitutions. (Cauchy.) 2. The order of every substitution group on n letters is a divisor of !. (Cauchy.) 3. All the substitutions which are common to two groups constitute a group. This is known as the cross-cut of these two groups. 4. The number of substitutions in a complete set of conjugate sub- 'stitutions under a group cannot exceed the quotient obtained by dividing the order of the group by the order of one of these substitutions. 5. The symmetric group of degree n, n>2, does not contain any invari- ant substitution besides the identity. 6. The alternating group of degree n, >3, does not contain an invariant substitution besides the identity. 7. Find the following products: abcdeXbdc= , ac-bdxbcfe= , "^ adeXabcXcicbed= , acXcdXdc 24 SUBSTITUTION GROUPS ICh. II 8. Find a value of / in the form of a product of transpx)sitions such that abcdefg =aeXafXagXbftXekXcdXt. 9. Transform ace-gf by hef, and also by ab-cd. 10. Write the 12 positive substitutions on four letters, and find the groups composed of all the substitutions on these four letters which transform into itself each of these 12 positive substitutions. 10. Co-sets and Double Co-sets. From the arrangement of all the substitutions of a group in the form of a rectangle in which the first line is composed of all the substitutions of a subgroup (9), it results directly that, if H is any subgroup of the group G, all the substitutions of G can be written in the following form : * G=H-\-nt2-{- . . . -\-ntx. In this notation, Hta{a = 2, . . . , X) stands for all the products formed by multiplying every substitution of H into /. The sets of substitutions represented by Hta are called co-sets f of G as regards H. It is important to observe that / may be replaced by any one of the substitutions of the co-set to which it belongs without changing the co-set. In other words, if two such co-sets have one substitution in common they are identical co-sets. It is sometimes desirable to include the subgroup H among the co-sets as regards H. In this case, the given X sets are called the augmented co-sets of G as regards H. Unless the contrary is stated, it will be assumed that these co-sets are distinct, so that every substitution of G appears once and only once in the co-sets. The set of multiplying substitutions I2, . . . , t\ can be chosen in A^"' ways, h being the order of //. Instead of multiplying H on the right we could have mul- tiplied on the left. Hence all the substitutions of G can also be written in the form G=H+t'2H+ . . . -\-t\n. *This notation is due to Galois. t The concept of co-sets was used by E. Galois, but he did not use a special name for it. H. Weber used the term Nehengruppen to represent what we her call co-sets. The latter term seems not to have been used for this concepi before 1910, Quarterly Journal of Maihcmalks, vol. 41 (1910), p. 382. 10) CO-SETS AND DOUBLE CO-SETS 25 It will be proved later ( 33) that it is always possible to select the X 1 multipliers in such a manner that / = /, o = 2, . . ., X. By taking the inverses of each of the co-sets in the formula of the first paragraph of this section, it results that G^H-\-t2-'H-h . . . -\-tx-'H. Hence it is also possible to replace / by ta~^ in the preceding formula. If fl^i and H2 are any two subgroups of G, the symbol H\taH2 is called a double co-set * of G as regards Hi and 5^2. It implies that each of the substitutions oi Hxta is multiplied on the right by every substitution of 112. All of these products are repre- sented by the single symbol H\tjl2. WTiile all of the substi- tutions of a co-set are distinct, those of a double co-set need not be distinct. If one substitution of G occurs exactly k times in such a double co-set, every substitution of the double co-set occurs exactly k times among the products represented by this double co-set. We proceed to prove this statement. Consider the product of the two groups H1H2, that is, all the products obtained by multiplying every substitution of Hi on the right by all the substitutions of H2. If Hi and H2 have exactly p substitutions in common, it is clear that each substitution of Hi and each of H2 will appear exactly p times in H1H2. In fact, if h is any substitution of ^1, the p sub- stitutions obtained by multiplying /i on the right by the sub- stitutions which are comm.on to Hi and H-2 will yield the same products when they are multiplied on the right by Ho. As no other substitution of Hi can yield any of these products, being in the same co-set of G as regards H2, it results that the product H1H2 involves each one of its substitutions exactly * Double co-sets were first used by A. L. Cauchy, Paris Compies Rendus, vol. 22 (1846), p. 630. They were more fully devdopetl by Frobcnlus, CreUe, vol. 101 (1887), p. 273. The term double co-set was first used with this meaning in Bulletin of the American Mathematical Society, vol. 17 (1911), p. 292. 26 SUBSTITUTION GROUPS (Ch. II p times, p being the number of the substitutions which are common to Hi and H2. It is now very easy to find the number of times that a given substitution of the product HitJI-z appears in this product. In fact, HitH2 = tta~^IIitaH2, and hence Ilitjh is the product of / and two groups. If a substitution s occurs exactly k times in ta'^Hita-Ho, the substitution t^s must occur exactly k times in IIitaHz. That is, each of the substitutions of H\tjl2 occurs exactly k times in this double co-set, k being the number of the substitutions common to the two groups ta~^Hita and H2. The following special case of this theorem is often useful. The number of the distinct substitutions in the double co-set H\tjl2 is equal to the product of the orders of Hi and H2, divided by a number which is the common order of two subgroups contained in Hi and 7/2 respectively. The double co-set Hitjl2 is not changed if / is replaced by any one of the other substitutions of this double co-set, since the co-sets Hit^ is not altered when / is replaced by any other substitution of this co-set, and likewise for the co-set 4^2. Hence two such double co-sets of G have either no substitution in common or they are identical. The possible double co-sets of G as regards the two subgroups Hi and H2, in the given order, are therefore completely determined by these subgroups. If we count only the distinct substitutions of these double co-sets we may write G in the following form: G^Hi'H2-{-Hit2H2-\- . . -\-HityH2- Each substitution of G appears once and only once in the second member of this identity, if these symbols are used to represent only the distinct substitutions which occur among the possible products represented by the symbols. Double co-sets are commonly used with this restricted meaning, and we shall hereafter use them in this sense unless the contrary is stated. Hence two of these double co-sets do not necessarily represent the same number of substitutions, but these numbers are always in accord with the theorem expressed above. 111 SYLOW'S THEOREM 27 11. Sylow's Theorem.* The theorem of the preceding section may be used to prove another very fundamental theorem known as Sylow's theorem, which asserts that every group whose order is divisible by P"* but not by p"*'^^, p being a prime number, contains a subgroup of order />. Such a subgroup is called a Sylow subgroup of the group. E. Galois made the remark in his Manuscripts, published by Jules Tannery in 1908, page 39, that a group whose order is divisible by p contains a subgroup of order p. This theorem was proved by A. L. Cauchy in 1845. It was extended to the case mentioned above by L. Sylow, in an article published in the M athematische Annalen, volume 5, 1872, page 584. We shall establish this theorem, treating first the special case when the group G is symmetric and of prime power degree. It is evident that the symmetric group of degree p, p being any prime number, contains a subgroup of order p. As p\ is not divisible by p^ this establishes Sylow's theorem for this special case. We proceed to establish the theorem for the symmetric group G of degree p"'^^. This group contains the following substitution composed of p cycles: s = ai . . . Gp-Gp+i . . . a2p- . . . -apo+i-p+i . . . ajft+i. It contains also the subgroup H composed of all the substi- tutions on these ^"+^ letters which transform s into itself. We proceed to prove that H includes a Sylow subgroup of G. The subgroup H includes a subgroup A' composed of all the substitutions obtained by multiplying in every possible way all the substitutions generated by the separate cycles of s. Since each cycle is of order p and there are p"' cycles in J, it results that the order of i^ is These p" cycles are permuted under H according to the sym- riietric group G' of degree p" which transfonns A' into itself. As G' is of a smaller degree than G, and as the symmetric group of degree p contains a Sylow subgroup, we may assume, in a * A different proof of this theorem will be given in i 27. 28 SUBSTITUTION GROUPS [Ch. II proof by complete induction, that G' contains a Sylow subgroup of order />"*', where m =- -. p-\ Since the p cycles of j are transformed under E according to a group of order p^\ and since the substitutions of U which permute these cycles according to these various substitutions transform K into itself, it results that E involves a subgroup E2 of order pn' ,^ = pm^ where m = - As p^ is the highest power of p which divides the order of G, in accord with the well-known theorem in number theory which has just been used, it results that G contains a Sylow subgroup of order />** whenever G' contains such a subgroup of order p^'. That is, Sylow's theorem has been established, by the method of complete induction, for every symmetric group whose degree is a power of p. It is now easy to establish Sylow's theorem for every pos- sible substitution group. Let Hi be any such group. As the symmetric group of degree n includes the symmetric group of degree n 1, it results that it is possible to find a value for a such that J^i is contained in the symmetric group G of degree pf^^. Let H2 represent any Sylow subgroup of order /?"* con- tained in G, and write the substitutions of G in the form of double co-sets as regards the two subgroups Hi and H2, as follows: G=HiH2-\-Hit2H2-\- . . . -\-H1tyH2. Let p^ be the highest power of p which divides the order of Hi. The number of distinct substitutions in each of these double co-sets is a multiple of />"+^-^^*, according to a theorem of the preceding section. This number cannot be divisible by p'^'^^ for every one of these double co-sets, since the order * This result is contained in the well-known formula for the highest power of a prime which divides !. Cf. Encydopidic dcs Sciences Mathimaiiques, tome 1, vol. 3, p. 4; K. I). Carmichacl, Theory of Numbers, 1914, p. 26. 11] SYLOW'8 THEOREM 29 of G is not divisible by />'""'" \ Hence there must be at least one of these double co-sets in which 6 = /3. As />* is the order of a subgroup of Hi, it follows that Hi must contain a Sylow subgroup. Hence every possible substitution group contains at least one Sylow subgroup corresponding to every prime number which divides the order of the group. If Hi contains more than one subgroup of order fP, let ^1, K2, , ^x represent all its subgroups of this order, A substitution s of Ki which is not also in K2 cannot transform A'2 into itself, otherwise 5 and K2 would generate a group whose order would be divisible by />^+^ Hence the substitutions of A'l must transform K2 into a complete set of conjugates under A'l and this set contains />" of these X subgroups. If p^' is less than X 1, this process can be repeated until all of these groups are exhausted. It is therefore necessary that X 1 be divisible by p. That is, the number of the Sylow subgroups of order p^ contained in any group is always of the form \+kp. In other words, this mmiber is always =l(mod p). From this theorem it results directly that every subgroup of order p"^ of a group G is contained in a Sylow subgroup of G, or is itself a Sylow subgroup of G. In fact, such a subgroup must transform into itself at least one of the Sylow subgroups of G. If it were not contained in this Sylow subgroup, G would involve a subgroup whose order would be a higher power of p than the order of its Sylow subgroup. It is also clear that every operator of order />* which is in G and transforms into itself a Sylow subgroup of order ^ must be contained in this Sylow subgroup. Another important elementary result in regard to the Sylow subgroups should be observed here. Suppose that the given X Sylow subgroups were such that they could not all be trans- formed into each other by the substitutions of these subgroups. There would therefore be a set of // which would be transformed only among themselves by all these substitutions. By trans- 30 SUBSTITUTION GROUPS (Ch. II forming these h subgroups by the substitutions of one of their own number it would result that /f would be of the form \-\-k\p, .and by transforming the same subgroups by the substitutions of a subgroup of order ^ which is not included among these // subgroups, it would result that h would be divisible by p. Hence all the Sylow subgroups of order p^ in any substitution group constitute a single complete set of conjugates under this group. In fact, they constitute such a complete set under these Sylow subgroups. The three results: that Sylow subgroups of order ^ always exist, that their number is of the form \-\-kp, and that they form a single set of conjugates under the group, are very closely related. Generally these three results are implied by the expression " Sylow's theorem." All of them are of fundamental importance. In fact, if the theorems of group theory were arranged in order of their importance Sylow's theorem might reasonably occupy the second place coming next to Lagrange's theorem in such an arrangement. EXERCISES 1. By means of Sylow's theorem prove that every substitution group of order 20 contains only one subgroup of order 5, and either one or five subgroups of order 4. 2. Every group of order pq, p and q being distinct primes and P>q, contains only one subgroup of order p. 3. A group of order 15 contains only one subgroup of each of the orders 3 and 5. Hence a group of order 15 is cyclic. 4. Find two substitutions of order 2 such that their product is of order 7, and verify that they generate a substitution group of order 14. 5. The number of subgroups of order p in the symmetric group of degree p is {p2) !, p being any prime number. Hence (^ 2) !s i(mod p). 6. The number of the subgroups which are generated by cyclic sub- stitutions of order n and arc contained in the symmetric group of degree . ( 1)! , X , . , . , n IS , 0() bemg the totient of n. 7. The symmetric group of degree n, n>3, contains more than one Sylow subgroup of order ff, whenever n! is divisible by p. 12. Transitive Groups, and Average Number of Letters in its Substitutions. The two substitution groups of order 4 1, ab'cd, ac'bd, ad'bc, 1, ab, cd, ab-cd 121 TRANSITIVE SUBSTITUTION GROUPS 31 represent two very important types of groups. In the former each letter of the group is replaced by every other letter by the various substitutions of the group. Such a group is said to be transitive. In the latter of these two groups, there is a letter which is not replaced by every other letter, and hence this group is called intransitive. Every substitution group evidently belongs to one and only one of these two t^-pes. Every symmetric group and every alternating group is transitive. Suppose that G is a transitive group on the n letters ai, a2, . . . , On. There is at least one substitution in G which does not involve the letter ai, viz., the identity. In general, the substitutions of G which omit ai constitute a subgroup Gi of order gi. As G is transitive it must involve a substitu- tion which replaces ai by a2, and this substitution transforms Gi into G2, G2 being composed of all the substitutions of G which omit 02. Hence G contains n conjugate subgroups Gi, G2, . . . , Gn, each of which is composed of all the substitutions of G which omit a letter. It is not necessary that all of these n subgroups be distinct. In fact, in the octic group they form two pairs of identical subgroups. All the substitutions of G can be arranged in a rectangle as follows, the substitutions of Gi forming the first row: 1, S2, SS, . , Sgi h, S2t2, S3t2, . . ., Sg^h h, S2t3, 53/3, . . ., Sg^3 t\, S2t\, Satx, . . ., Sg^t\ where \=g/gi. If ^2 replaces ai by 02 then all the ^i substi- tutions of the row involving /2 have this property. If any other substitution of G should replace ai by 02, all the distinct prod- ucts, obtained by multiplying its inverse into itself and into all the substitutions of the row involving /a, would transform 02 into itself. As this would give more than gi distinct sub- stitutions, the row which involves /2 contains all the substitu- tions of G which replace ai by 02. 32 SUBSTITUTION GROUPS [Ch. II Since similar remarks apply to every other row, it results that X = . That is, the order of tlie subgroup formed by all the substitutions of a transitive group which omit a given letter is equal to the order of the group divided by its degree. From the given rectangle it follows that each letter occurs gi{n l) times in the substitutions of G. Hence these substitutions involve gin{ni) =g(w 1) letters. That is, the average number of letters in the substitutions of a transitive group is equal to the degree of the group diminished by unity* In particular, the average number of letters in all the possible substitutions on n letters is n\, and this is also the average number in all the positive substitutions on these letters when n> 2. 13. Intransitive Substitution Groups. One of the simplest examples of an intransitive substitution group may be obtained by multiplying together transpositions on distinct sets of let- ters. For instance, the intransitive group generated by the following three transpositions, ab, cd, ef, is of order eight and contains seven substitutions of order 2, besides the identity. This is a special case of the elementary theorem which afl&rms that h transpositions on // distinct pairs of letters generate a group of order 2^ and of degree 2//. This group is intransitive when h>\. By multiplying all the substitutions of any transitive group by all those of another transitive group, represented on a dis- tinct set of letters, there results an intransitive group whose order is the product of the orders of these two transitive groups, and whose degree is the sum of their degrees. Hence it is clear that it is possible to construct an unlimited number of different groups from any given group by representing the group on distinct sets of letters and then multiplying the sub- stitutions in every possible manner. Groups obtained in this manner are sometimes called powers of the given group, an index being used to indicate the number of times the group was used * This interesting theorem was given explicitly for the first time by G. Frobenius, CreUe, vol. 101 (1887), p. 287. 13] INTRANSITIVE SUBSTITUTION GROUPS 33 as a factor. The groups considered in the preceding para- graph constitute a special class of such powers. Another way of forming an unlimited number of non-con- jugate substitution groups from a given substitution group is by a process called establishing a (1, 1) correspondence or a simple isomorphism. For instance, the following three sub- stitution groups of order 2: 1, ab-cd; 1, ab-cd-ef; 1, ab-cd-ef-gh are obtained by establishing simple isomorphisms between the groups 1, ab and 1, cd; I, ab, I, cd, and 1, ef; 1, aft, 1, cd, 1, ef, and 1, gh, respectively. It is clear that we can establish such simple isomorphisms between any number of groups, obtained by writing a given transitive group on distinct sets of letters. All the groups thus obtained are merely different ways of representing the same group of order 2 considered abstractly, and these isomorphisms show that there is no upper limit to the number of letters of the substitution groups which represent such a group. The two given methods of constructing intransitive sub- stitution groups are called the direct prodiict method and the simple isomorphism method. They are the simplest methods for constructing such groups and the other possible methods are based upon them.* Before entering upon a consideration of other methods it should be observed that every intransitive group is composed of transitive constituent groups, and that it can be constructed by establishing some correspondence between these constituent groups. Let G be any intransitive group, and consider the letters which replace a given letter, a in the substitutions of this group. If a is replaced by b in some substitution si and there is also a substitution 52 in which b is replaced by c, then there must be a third substitution 53 = siS2 in which a is replaced by c. Hence a and all the letters by which a is replaced constitute the letters of a transitive constituent group. If 1, S2, . . . , 5* represent the substitutions of this con- * Cf. Bolza, American Journal of Mathtmatks, vol, 11 (1), p. 185. 34 SUBSTITUTION GROUPS (Ch. II stituent group K, then each of these k substitutions is found in the same number of the substitutions of G, and hence ^ is a divisor of g, where g is the order of G. In the simple isomorph- ism method, k=g. All the substitutions of G which involve only the identity from K constitute an invariant subgroup H of G, and A' is said to be a quotient group of G as regards H. This quotient group is commonly represented by the following symbol: * G/H=K. For instance, consider the following intransitive group G: 1, cde, ced, ab-cd, ah -de, ah-ce. One of the transitive constituent groups is 1, ab = K, and the invariant subgroup of G, which involves only the identity of this constituent group, is as follows: E = l, cde, ced. The quotient group G/H=K may also be regarded as a group in which the three substitutions of H are regarded as one sub- stitution, while the remaining three substitutions constitute the other substitution. The other transitive constituent of G is simply isomorphic with G, and the given correspondence is sometimes represented by the following symbol: 1 cde 1 ced cd de ab ce As an instance of a more general correspondence, we may consider the group of order 18 composed of all the positive substitutions in the direct product of the symmetric group of degree 3 represented on two distinct sets of letters. This group is represented as follows: This symbol was used by C. Jordan, Bulletin de la Sociili MaiMmatique de France, vol. 1 (1K72), p. 46. The symbol is often credited to Holder, who used it in 1889. Cf. H, Weber, Kleines Lehrhuch dcr Algebra, 1912, p. 192. 14] SUBSTITUTIONS INVARIANT UNDER A GROUP 35 1 1 abc dej acb dfe ah de he ef ac df If we let K represent the former of these two constituent groups it is clear that H=l, def, dfe, and that G/n=K can be regarded as a group in which three substitutions of G are con- sidered as a single substitution. 14. Substitutions which are Commutative with Each of the Substitutions of a Transitive Group. When Gi, the subgroup composed of all the substitutions which omit a given letter, reduces to the identity, the transitive group G is said to be a regular group. Regular groups are especially important since every possible group of finite order can be represented as a regular substitution group, as will be proved later. We pro- ceed to prove the important theorem, which we shall call Jordan's theorem* that the total number of substitutions on n letters which are commutative with every substitution of a regular group G on the same n letters constitutes a group C which is conjugate or similar to G; i.e., it can be transformed into G by some substitution. This theorem results ahnost immediately if we multiply all the substitutions of (7=1, ^2, S3, . . ' , Sg successively on the right and on the left as follows: t G = * This theorem was first proved in Jordan's thesis, Paris, 1860, p. 39. t The two simply isomorphic regular substitution groups which are repre- sented by these square arrays have been called potential and antipotential groups respectively. G. FrattinI, Aui della R. Accademia del Uncei. Memoria, vol. 14 (1883), p. 144. 1, S2, S3, . . ., Sg 1, S2, S3, . . ., Sg ^2, S2^, S3S2, ., SfSo ^2, 52^ 52^3, . . ., SiSg ^3, S2S3, S3^, . . ., SgS3 G' = ^3, Sg, ^3^2, S9S2, ^3^ . V3, . . ., S3Sg Sg, S2Sg, SaSg, r 2 . ., Sg . ., V' 36 SUBSTITUTION GROUPS [Ch. II The permutation of the substitutions of each of these rows as regards the first row represents a substitution, and we may assume, without loss of generality, that the substitutions repre- sented by the first square, when the various rows are associated successively with the first row, are the substitutions of G. Any one of these rows, say the one involving Sa, represents a substi- tution obtained by multiplying all the substitutions of G on the right by Sa', while any row of G', say the one involving s^, represents a substitution obtained by multiplying all the sub- stitutions of G on the left by Sp. Since we get the same result when we multiply all the substitutions of G first on the right by Sa and then on the left by Sp, as when we multiply them first on the left by Sp and then on the right by Sa, as a consequence of the associative law, it follows that each substitution of G is commutative with every substitution of G', and vice versa. Moreover, G' includes all the substitutions on these letters, which are commutative with every substitution of G, since every such substitution must involve all the letters of G and the totality of these substitutions forms a group. As G and G' are different ways of representing the group G they must be simply isomorphic. It remains to prove that they are conjugate. If we establish a simple isomorphism between G and G' in such a way that all the substitutions begin with the same letter, the second letters in all the substitutions of these groups represent the substitution by means of which G may be transformed so that the first two letters in each one of its substitutions are the same as the first two letters of the corresponding substitution in G'. Since this transfor- mation will lead to simply isomorphic groups, and since two simply isomorphic regular groups have the property that the corresponding substitutions are identical whenever the first two letters of all these substitutions are the same, we have proved Jordan's theorem; viz., with every regular group of order n there is associated another regular group of order n such tltat each of tliese groups is composed of the total number of substi- tutions on these n letters which are commutative with every substi- tution of the other group. 14] SUBSTITUTIONS INVARIANT UNDER A GROUP 37 The two groups G and G' whicli are defined by Jordan's theorem are called conjoints. When G is abelian it coincides with its conjoint and vice versa. A special case of this theorem relating to a cyclic group was proved above in 8. Suppose that G is transitive and of degree n, but not regular, and that all the substitutions of G which omit ai omit also a2, . . . , Qa- Hence Gi, which is composed of all the substitutions of G which omit ai, is transformed into itself by all the substitutions of G which replace ai by a2, . . . , That is, Gi is trans- formed into itself by a subgroup H of order agi. All of the substitutions of H, except those of Gi, must involve each of the letters ai, a2, . . . , da- Hence H is an intransitive group, and the components of its substitutions involving the letters ai, 02, . . . , Oa must form a regular group. This regular group is a transitive constituent of H and the subgroup Gi, which is transformed into itself by H, corresponds to the iden- tity of this transitive constituent. Let Ci be the conjoint of this constituent, and consider the n/a transforms of Ci under G. A (1, 1) correspondence can be established between the substitutions of these n/a transforms such that each substitu- tion is of degree n and is transformed into itself by G. To do this it is only necessary to regard as one substitution all the transforms of a single substitution of Ci. This proves a theorem due to H. W. Kuhn,* which is a generalization of Jordan's theorem and may be stated as follows: A necessary and sufficient condition that there are a substi- tutions on the letters of a transitive group, which are commuta- tive with every substitution of the group, is that its subgroup which is composed of all its substitutions which omit one letter omits exactly a letters. If Gi is of degree 1, the identity is therefore the only substitution on these n letters which is commutative with every substitution of G, as is also otherwise evident. * American Journal of Mathematics, vol. 26 (1904), p. 67. 38 SUBSTITUTION GROUPS [Ch. 11 EXERCISES 1. Every transitive group of degree n involves at least n 1 substitu- tions which separately involve all the letters of the group; if it contains also substitutions of degree na, !<<, it must contain more than 1 substitutions of degree n. Suggestion. The average number of letters in the substitutions is n 1. 2. If all the substitutions which omit a given letter of a transitive group of degree n constitute a group of degree nl, then the n conjugates of the latter are transformed under the transitive group in exactly the same way as its letters are transformed. 3. A transitive group composed of invariant substitutions is necessarily regular. 4. If the order of a transitive group is p^, p being a prime number, the subgroup composed of all its substitutions which omit a given letter omits />"* letters, where mo>0. 5. All the substitutions of highest degree (n) in any transitive group generate a transitive group of degree w, which either coincides with the original group or difTers from it merely with regard to substitutions of degree 1. Suggestion. Use the theorem that the average number of letters in all the substitutions of a transitive group of degree n is nl, while in an intransitive group this number is smaller. 6. The total number of the substitutions which are conjugate to a given substitution of a group must generate either the entire group or an invariant subgroup. This theorem remains true if the word conjugate is replaced by the word similar. 7. The number of the substitutions of degree p^ and of order p in the symmetric group of degree n is prime to p whenever p^ is the highest power of P which does not exceed n. Cf. Annals of Mathematics, vol. 16 (1915), p. 169. 15. Primitive and Imprimitive Groups. When the sub- group Gi, composed of all the substitutions of a transitive group G which omit a given letter, is of degree na, n being the degree of G, we have seen that the letters of G may be divided into n/a sets, each set involving a distinct letters, such that these sets are transformed as units by all the substitutions of G. Such a substitution group is said to be imprimitive whenever o> 1, and the sets of a letters are called its systems oj imprimitivily. These systems of imprimitivity are trans- formed under G according to a transitive group which has a 15] PRIMITIVE AND IMPRIMITIVE GROUPS 39 (1, w) correspondence with G. If m> 1, G must therefore con- tain an invariant subgroup of order m which transforms each of these systems of imprimitivity into itself. If m = l, each substitution of G, besides the identity, must transform at least one of these systems of imprimitivity into another. While every transitive group whose Gi omits more than one letter is necessarily imprimitive, it does not follow that the Gi of an imprimitive group must omit more than one letter. We proceed to prove that a necessary and sufficient condition that G is imprimitive is that Gi is contained in a larger subgroup of G. In other words, a necessary and sufficient condition that G is imprimitive is that Gi is a non-maximal subgroup of G. It should be observed that this theorem connects the theory of imprimitivity with the theory of abstract groups. Every transitive substitution group which is not imprimitive is said to be primitive. The given theorem is contained in a more general theorem which may be stated as follows: A necessary and sufficient condition that a complete set of conjugate substitutions or sub- groups of G is transformed under G according to an imprimitive substitution group is that the largest subgroup of G which trans- forms into itself one of these substitutions or subgroups is contained in a larger subgroup of G. That this condition is sufficient results from the fact that if this largest subgroup K, which transforms into itself a substitution or subgroup L, is contained in a larger subgroup H, then the number of the conjugates of L under H is equal to the quotient obtained by dividing the order of H by the order of K, and H involves all the substi- tutions of G which transform these conjugates among them- selves. Every substitution of G which is not in 77 must therefore transform the set of conjugates of L under // into an entirely new set of conjugates, and therefore this set of conjugates is transformed as a unit. Hence all the conjugates of L can be divided into sets such that they are all transformed as units and such that no two sets have a common substitution or sub- group. On the other hand, if the complete set of conjugates 40 SUBSTITUTION GROUPS [Ch. li to which L belongs is transformed according to an imprimitive group there must be such a group as H which includes A', since all the substitutions which transform among themselves the substitutions or subgroups, corresponding to a system of im- primitivity, must constitute a group. Hence the given general theorem is established. To deduce the special case relating to the primitivity of G when Gi is of degree w 1, we have only to observe that a necessary and sufficient condition that G\ is transformed into itself by only its own substitutions, is that its degree is w 1. As G transforms the conjugates of G\ in exactly the same way as it transforms its letters, whenever the degree of Gi is w 1, and as we assume in the present case that the degree of Gi is exactly 1, it results directly from the given theorem that G transforms the conjugates of Gi, and hence also its letters, according to a primitive group^ whenever Gi is maximal, and only then. If G\ is a transitive group on 1 letters, G is said to be doubly transitive, and every pair of letters of G is transformed into every other pair by the substitutions of G. In general, G is said to be r-Jold transitive, whenever Gi is (r l)-fold trans- itive and of degree w 1. Since the order of a transitive group is always a multiple of its degree, it results that the order of an r-fold transitive group is a multiple of n{n\) . . . (n r+l). A group which is more than simply transitive is said to be multiply transitive. The alternating group of degree n is ( 2)-fold transitive and the symmetric group of degree n is said to be either -fold or ( l)-fold transitive. We shall generally say that this group is (w l)-fold transitive. With the exception of the alternating and the symmetric groups no group is known which is more than five-fold transitive and only two such five-fold transitive groups are known. These groups are of degrees 12 and 24 respectively and were dis- covered by E. Mathieu in 1861. The theory of multiply transitive groups has not yet been extensively developed, and it seems to offer great difficulties. 161 GKOUFS ON FOUR LE1TEU8 41 EXERCISES 1. If all the substitutions of order w, m>2, in a group are conjugate, they must be transformed under the group according to an imprimitive group. Suggestion: The generating substitutions of one cyclic subgroup are transformed into all of those of another. 2. If an imprimitive group contains substitutions besides the identity which do not interchange any of its systems of imprimitivity, in a given set of systems of imprimitivity, all such substitutions constitute an invari- ant subgroup. 3. Every regular group of composite order is imprimitive, and involves as many different sets of systems as it has subgroups, excluding the identity. 4. A transitive group of order p", p being a prime number and o>l, is always imprimitive. Suggestion: Consider its complete sets of congjuate substitutions and observe that each of these sets involves p^ distinct substitutions. 5. Every invariant subgroup besides the identity of a primitive group is transitive. 6. The total number of substitutions which are commutative with every substitution of the intransitive group obtained by establishing a simple isomorphism between n, n>2, symmetric groups of degree n, written in distinct sets of letters, constitute a conjugate intransitive group. 16. Groups Involving no More than Four Letters. The only possible substitution group on two letters is 1, ah. Since every system of intransitivity must involve at least two letters, a group of degree 3 is necessarily transitive. It is also included in {abc)all* and hence its order is a divisor of 6. Therefore {ahc)all and {ahc) are the only two possible groups of degree 3. If a group of degree 4 is intransitive, each of its two systems of intransitivity must be of degree 2. The largest intransitive group of degree 4 is therefore the direct product of {ah), (cd). The other possible intransitive group is a simple isomorphism between the substitutions of these transitive groups of degree 2. Hence there are two and only two intransitive groups of degree 4; * The notation (aioj . . . an)aU is used to represent the symmetric group of degree n, while (aiOj . . . On) represents the group generated by the cyclic sub- stitution fliOj ... On. It should, however, be emphasized that the symbol (aiOi ... On) is also often employed to denote the substitution diai . . . ii,. Since there is no uniformity of usage along this line, the reader is frequently obliged to determine the meaning from the context. 42 SUBSTITUTION GROUPS [Ch. II one of these is of order 4 while the other is of order 2. Their substitutions are as follows: 1, ab, cd, ab-cd; 1, ab-cd. According to Sylow's theorem, (abed) all involves at least one subgroup of order 8 and all its subgroups of this order are conjugate. Since all conjugate groups are regarded as identical in the enumeration of groups, there is one and only one sub- stitution group of degree 4 and of order 8. This is known to be transitive (2). We shall represent it by the symbol {abcd)B. Since the order of every transitive group is a mul- tiple of its degree, and since a group must be included in the symmetric group of its own degree, it results that the order of a transitive group of degree four is 4, 8, 12 or 24. As there is one and only one such group of each of the orders 8 and 24, it remains to determine all the possible groups of orders 4 and 12. We know that there are two transitive groups of the former order; viz., the subgroups of the octic group con- sidered in 2, and there is one group of the latter order; viz., {abcd)pos* We proceed to prove that no other groups of these orders are possible. Another transitive group of order 4 would also be regular, since the average number of letters in its sub- stitutions is 3. It could not be cyclic, since there is one and only one cyclic group of each order, and two simply isomorphic regular groups are conjugate. If it were non-cyclic it would involve all the possible substitutions of the form ab-cd in the symmetric group of degree 4. This proves that there are only two regular groups of degree 4. One of these has three con- jugates under the symmetric group while the other is invariant under this group. To prove that there is only one group of order 12 and degree 4, we observe that every such group would have to contain a subgroup of order 3 according to Sylow's theorem. As all the subgroups of order 3 in {abcd)aU are conjugate, we may assume that every group of order 12 and degree 4 includes iflbc). Since this subgroup could not be invariant under a tran- * The symbol (aioj . . . an)pos represents the alternating group of degree n. 17] SIMPLICITY OF THE ALTERNATING GROUP 43 sitive group of degree 4, every group of this degree and of order 12 must involve the 4 subgroups of order 3 in {abcd)all, and hence it must be identical with {ahcd)pos. That is, there are exactly five transitive groups of degree 4, one of each of the orders 24, 12, 8, and two of irder 4. Each of these possible transitive groups has at least two substitutions in common with the non- cyclic regular group. It has been observed that (abcd)pos cannot involve a sub- group of order 6, since its Sylow subgroups of order 3 are trans- formed into themselves by their own substitutions only under {abcd)pos. We have here an instance where a group whose order is divisible by 6 does not contain a group of order 6. That is, while the order of every subgroup is a divisor of the order of the group there is not neccessarily a subgroup for each divisor of the order of a group. The ten groups whose degrees do not exceed 4 may be replresented as follows: * (a6), {abc)all, (abc), {abcd)all, {abcd)pos, {abed) 8, (abed), (abed) 4, iab)(cd), (ab-cd). 17. Simplicity of the Alternating Group of Degree n, n?^4. If a group does not contain any invariant subgroup besides the identity it is said to be a simple group. All other groups are said to be composite. We proceed to prove that the alter- nating group is simple except when its degree is 4. Suppose that si, 52 are two cyclic substitutions of order greater than 2, such that one can be obtained from the other by interchanging two adjacent letters. That is, si= ... ayOaOfi . . . , 52= . . . ayOfiaa . . . Hence SiS2~^=at^yap. That is, if the order of a substitution exceeds 2 it is always possible to find anotJier substitution similar to it such that the product of the two is a cycle of order 3. By means of this elementary theorem and the fact that every posi- * A fundamental problem of substitution groups is the determination of all the substitution groups of degree w. This problem has been completely solved when n does not exceed 11. The noted French mathemalidan .\. L. Couchy was the first to do serious work along this line; Paris Comptes Rendus, vol. 21 (184.5), p. 136a. 44 SUBSTITUTION GROUPS [Ch. II tive substitution is the product of cycles of order 3, it is easy to prove that every alternating group whose degree exceeds 4 is simple. Assume that G is the alternating group of degree , n>4, and that it involves an invariant subgroup H. As H involves all the substitutions of G, which are conjugate with any one of its substitutions under G, it cannot involve any substitutions of order 2 without also containing substitutions whose orders exceed 2. This results immediately from the fact that the positive substitution ah-cd . . . of order 2 is transformed by hde into a substitution which is not commutative with it, and hence the product of these two substitutions has an order which exceeds 2. If H is not the identity, it must therefore involve either substitutions of an odd prime order, or substitutions involving cycles of even order greater than 2. As the cyclic groups generated by such "feubstitutions are evidently transformed into themselves by negative substitutions, these cyclic groups are transformed into all their conjugates under the symmetric group of degree n by substitutions of G. That is, H must involve all these conjugates, and hence it must involve all the substitutions of the form abc in G. In other words, H coincides with G whenever it exceeds the identity. This proves the important theorem: the alternating group of degree n is simple whenever w>4. The alternating group of degree 3 is evidently simple, but the alternating group of degree 4 contains 1, ah-cd, ac-hd, ad-bc a.s an invariant subgroup, as was observed above. Hence every alternating group, with the exception of the alternating group of degree 4, is simple. In this special case the alternating group contains a group of order 4 as an invariant subgroup. From what precedes, it results immediately that the s>Tnmetric group of degree w contains only one subgroup of order !/2. If it contained two such subgroups, one would contain only !/4 positive substitutions, and these would constitute a sub- group of the alternating group. This is impossible, since such a subgroup would contain all the substitutions of odd order and hence all the substitutions of the form abc, 18] GROUPS OF DEGREE FIVE 46 18. Groups of Degree Five. The intransitive groups on five letters must involve transitive constituents of degrees 3 and 2, and hence all of them are contained in the direct product of {abc)all, (de). Consequently there are three such groups one of order 12 and two of order 6. They may be represented as follows: {abc)all(de), {abc)(de), \{abc)all(de)\pos. AU the transitive groups of degree 5 are primitive, and they involve either only one, or six subgroups of order 5. As these six groups must generate the alternating group, according to the preceding section, it follows that the alternating and the symmetric groups of degree 5 are the only ones which involve six subgroups of order 5. If such a group contains only one subgroup of order 5, its order divides 20, since a generating substitution of the subgroup of order 5 is transformed into itseK by only five substitutions on these 5 letters. We may evidently suppose that each of the possible groups involves the same subgroup of order 5 since all these subgroups are con- jugate. Hence there is only one such group of each of the orders 20, 10, 5. The five transitive groups of degree 5 may be repre- sented as follows: (abcde), (abcde) 10, (abcde)20, (abode) pos, {abcde)all. The group (abcde) pos is especially interesting since it is the simple group which has the smallest possible comixjsite order. This group is simply isomorphic with the total number of movements which transform the icosahedron into itself, as we shall see later, and hence it is frequently called the icosa- hedron group. Galois was the first to observe that it is simple, and he also observed that it is the smallest simple group of composite order. Sir W. R. Hamilton observed, in 1856, that this group may be defined abstractly as the group generated by two substitutions of orders 2 and 3 respectively whose product is of order 5. That is, every pair of substitutions which satisfy these conditions generates a group of order 00, which is simply isomoqihic with (abcde)pos. 46 SUBSTITUTION GROUPS [Ch. II 19. Holomorph of a Regular Group. If G is a regular group of order n, all the substitutions on these n letters which trans- form G into itself constitute a group which has been called the holomorph * of G. For instance, the symmetric group of degree 3 is the holomorph of its subgroup of order 3, {ahcd)^ is the holomorph of {abed), and {abcd)20 is the holomorph of {abode). The holomorph of G includes the conjoint of G, and hence it is also the holomorph of this conjoint. If this conjoint is not identical with G, it is conjugate with G under a substitu- tion of order 2 which transforms the holomorph of G into itself. This substitution and G generate a group known as the double holomorph of G. Every non-abelian group has a double holomorph. Since the holomorph K ol G involves exactly n substitutions which are commutative with every substitution of G, it must transform the substitutions of G in k/n different ways, k being the order of K. The largest subgroup of degree n\ con- tained in K must also transform the substitutions of G in k/n different ways. This subgroup is known as the group oj isomorphisms of G.f Hence the order of the holomorph of a group is the product of the order of the group and the order of its group of isomorphisms. Since any two simply isomorphic regular groups are conjugate, the group of isomorphisms of G transforms G into every possible simple isomorphism with itself. That is, it transforms it into all its possible automorphisms. If a group involves substitutions which transform it into every possible automorphism, but does not contain any invari- ant substitution besides the identity, it is said to be a complete group. The symmetric group of degree 3 is evidently a com- plete group. The holomorph of a complete group is the product of the group and its conjoint. It is easy to prove that the The concept of holomorph was used by many early writers, but the term was introduced by W. Burnside in the first edition of his Theory of Groups, 1897, p. 228. t The statement relating to this matter in the EncyklopUdie der Mathemalischen Wissenschaften, vol. 1, p. 221, note 103, is inaccurate. The group of isomorphisms is one of the most important and also one of the most far-reaching concepts in group theory. 20] CLASS OF A SUBSTITUTION GROUP 47 symmetric group of degree 4 is also complete. In fact, this group is generated by any two of its cyclic subgroups of order 4. One of the generators of such a cyclic group can be selected in six different ways, and after it has been selected, a generator of the second cyclic group can be selected in four different ways. The two generators can therefore be selected in 24 different ways. Since (abcd)all contains no invariant substitution besides the identity, it must transform its own substitutions in these 24 possible different ways. Hence the holomorph of (abed) all is the direct product of this group and its conjoint. The order of this holomorph is therefore 576. EXERCISES 1. The symmetric group of degree n does not contain any subgroup of index p, if p is greater than 2 but less than the largest prime factor of n. (Cauchy, 1815.) Suggestion: Such a subgroup could not involve all the substitutions whose order is this prime factor, since these substitutions would generate the alternating group. 2. Prove that the symmetric group of degree n is generated by a cyclic substitution of degree n1 and a transposition which connects any one of these letters with the remaining letter. 3. The group of order 48 on Xi, . . ., Xi, which transforms the func- tion XiXi+xsXt+XiXi into itself is imprimitive and involves an invariant subgroup of order 8. 4. The polynomial {xi+(t)Xi+ 0)^X3+ . . . +c/~^XrY is transformed into itself by the cychc group of order r on these variables, if w represents a number whose powers give all the rth roots of unity. Suggestion: If we multiply the polynomial within the parenthesis, which is known as the Lagrangian resolvent, by a power of u, we inter- change the variables cyclically. 5. Construct all the possible substitution groups that can be repre- sented on six letters. Cf. American Journal of Mathematics, vol. 21 (1899), page 327. 20. Class of a Substitution Group. A substitution group G of degree n is said to be of class na, a<n, if it contains at least one substitution of degree na but does not contain any substitutions besides the identity whose degree is less than n-a. For instance, every symmetric group is of class 2 while every alternating group is of class 3. It is easy to prove that 48 SUBSTITUTION GROUPS (Ch. II these two infinite systems are composed of all the possible primitive groups which are of class 2 and class 3 respectively. A substitution which actually contains exactly k letters is some- times said to be of class k. To prove that a primitive group G which is of class 2 and of degree n is symmetric, it is only necessary to observe that such a group contains at least two transpositions having a letter in common, otherwise G would be imprimitive. Hence G involves the symmetric group of degree 3. It must therefore contain two such symmetric groups which have two letters in common. Hence G contains the symmetric group of degree 4. By continuing this process, it results that G is the symmetric group of degree n. In exactly the same manner, it can be proved that if a primitive group of degree n contains a substitution of the form ahc it includes the alternating group of degree n. That is, if a primitive substitution group involves a transposition it is a symmetric group, and if it involves a substitution of the form abc without also involving a transposition, it is an alternating group. Suppose that G is a primitive group which contains a sub- stitution of degree and of order p, p>S, and that G is of degree n, n>p, p being a prime number. There must be at least two substitutions in G which are of degree and of order p, and which have some but not all their letters in common. Let 5i and 52 be two such substitutions. If S2 involves more than one letter which is not also contained in si, there is some power of Si in which two such letters are adjacent. The transform of S\ by this power will then be a substitution which has more letters in common with s\ than 52 has; but this trans- form involves at least one letter which is not contained in si. Hence we may assume that G contains two substitutions of degree p and of order p such that these substitutions contain exactly p\ common letters. These two substitutions generate a doubly transitive group of degree p-\-\. By continuing these considerations it results that if a primitive group of degree n contains a substitution of degree p and of order p, p being any prime number, this primitive group is at least {n p-{-\)-fold transitive. 20] CLASS OF A SUBSTITUTION GROUP 49 It is now easy to prove that a primitive group of class ^, />>3, cannot have a degree which exceeds />+2. In fact, if such a group were of degree />+3, it would be at least four- fold, or four times, transitive. Since any set of four letters of such a group can be replaced by an arbitrary set of four let- ters, it results that any four times transitive group must involve an intransitive subgroup H which has the sjTnmetric group of degree 3 for one constituent, and a transitive group on the remaining letters for the other constituent. In the present case the latter group is of degree p. In any transitive group of degree />, the subgroups of order p generate a simple group, since an invariant subgroup of a primitive group is transitive. If this simple group is not the entire group, it must be invariant under the entire group and the corresponding quotient group must be cyclic, since it is a subgroup of the group of isomorphisms of a group of order p, and this group of isomorphisms is cyclic, since p has primitive roots. Hence it results that H includes substitutions of the form ahc whenever G is of degree />+3, since its transitive con- stituent of degree p cannot give rise to a quotient group which is simply isomorphic with the symmetric group of degree 3 when />>3. That is, if a primitive group is of class p, p being a prime number greater than 3, the degree of this primitive group is at most p-\-2. There is evidently one and only one primitive group of degree p and of class />; viz., the group of order p. In order that a primitive group of degree />-fl be of class p, it is clearly necessary that this group be of order />(/>+l), and that it contain (/>+!)(/> 1) substitutions of order />. Hence p-\-\ must be of the form 2*", and if. p-\-\ is of this form there is one and only one such group. A primitive group of degree p-\-2 which is of class p must therefore include this primitive group of degree />+l. It must also contain a substitution of order 2 and of degree p-\-\ which transforms into its inverse one of the substitutions of order p in this group of degree />+ 1 . Hence there cannot be more than one primitive group of degree p-\-2 and of class />. 60 SUBSTITUTION GROUPS [Ch. II The fact that this primitive group actually exists was proved by C. Jordan,* but we shall not give the proof here. We have, however, established the following theorem: When the prime number p, p>S, is not of the form 2'*! there is one and only one primitive group of class p. When p is of the form 2* 1 there cannot be more than three primitive groups of class p. The study of primitive substitution groups by means of their classes was begun by C. Jordan, who proved that there is a finite number of such groups of every class. In recent years W. A. Manning has contributed new theorems on this interest- ing but difficult subject, t EXERCISES 1. If a primitive substitution group contains two transitive subgroups which can be transformed into each other by a transposition, the primitive group is alternating or symmetric. 2. Prove that the two substitutions abe cdf a.nd eg fc/ generate a group of order 168, and that this group is simple. 3. It is known that the simple group of order 504 can be represented as a transitive substitution group on 9 letters. Hence prove that it con- tains the abelian group of order 8 which is composed of seven substitu- tions of order 2 and the identity. * Journal de Mathtmatiques (2), vol. 17 (1872), p. 3.51. t W. A. Manning, Transactions of the American Mathemalical Society, vol. 11 (1912), p. 375. CHAPTER III FUNDAMENTAL DEFINITIONS AND THEOREMS OF ABSTRACT GROUPS 21. Introduction. In the preceding chapters we gave several examples of groups which were either in the form of substitution groups or could readily be associated with such groups. Some of the fundamental theorems of substitution groups were also developed. As the theory of groups is applic- able to many different subjects, it became a matter of economy of thought to develop and to state the main theorems of this theory in a language which is common to these various sub- jects. The efforts to accompHsh this end led gradually to what is known as abstract group theory. Many of the theorems proved in the preceding chapter can be used directly in the theory of abstract groups. In fact, we shall find that the theory of substitutions is a very useful means to study the abstract properties of groups. It has already been observed, 15, that the question of primitivity and imprimitivity of a substitution group has an abstract meaning, even if these terms apparently relate only to the notation of substitution groups. Some of the definitions used in the theory of substitution groups are not directly available for the study of abstract groups, and hence it will Be necessary to re-define some of the terms used in the preceding chapter. This is due to the fact that substitutions have inherent properties. For instance, the associative law is always satisfied when substitutions are multiplied, and hence we did not need to specify this law in defining a substitution group. On the contrary, this law should be included in a definition of an abstract group. 51 52 ABSTRACT GROUPS (Ch. Ill 22. Definition of an Abstract Group and a Few Properties of its Elements. In the theory of finite abstract groups we deal with a set of distinct symbols G=si, S2, . . . , Sg, and we assume that any two of them can be combined according to some law which is called multiplication, and which is denoted in the same way as multiplication is commonly denoted. This set of symbols represents a group provided the symbols satisfy the following conditions: 1. If any two of the three symbols in an equation of the form are contained in G, then the third is also contained in G, and it is completely determined by this equation. It is assumed that this statement includes the case when the two symbols which are contained in G are identically equal to each other. 2. The symbols of G obey the associative law. That is, iSaSfi)Sy=S{SpSy). From the former of these conditions it results that if one of the three s3niibols in SaSfi = Sy remains fixed while another assumes successively all the values of the symbols or elements of G, the third will also run through all these elements. In particular, there must be an element si such that SiSp=Sfi. Such an element is called the left-hand identity of Sp. This left-hand identity is the same for all the elements of G. To see this fact we multiply the last equation on the right by Sy, and then let Sy run through all the elements of G. In exactly the same way it may be observed that G contains only one right- hand identity s[. To prove that"5l=5i it may be observed that if we replace S0 by s[ in the last equation it results that SiSi = Si. 221 DEFINITION OF ABSTRACT GROUPS 53 Similarly, by letting Sa = si in the following equation it results that Sl5i=Si. That is, 5i = Sii in other words, every group contains one and only one element which does not alter the value of any element when it is used as a multiplier on the right or on the left. This element is called the identity of the group, and it is denoted by the symbol 1. Every other element changes all the elements into which or by which it is multiplied. Since g is finite there must be some finite power of ^^ (a = l, 2, . . , g) which is equal to some other power of this element. That is, the series 'a; ''a J 'a } j 'a must involve a term which is equal to a preceding term when n is taken sufficiently large. Let sj be the lowest power of s^ which satisfies the equation sj = sj'\ or sj'^^ = sj, 0<k<r. Since and since G involves only one identity, it results that sj=l= s^. That is, the lowest power of Sa which is equal to a lower power occurs after the identity appeared in the series of successive positive powers. On the other hand, since it results that r = ^+l. That is, the first power which is equal to a lower power in a series of successive powers of an clement is the one following the identity in this series. Hence the series is periodic and the number of different elements in each period is k. This number is called the order or period of s. Two powers of Sa whose exponents are of the form l+mk are equivalent whenever m is any positive integer or zero. Negative exponents are introduced by assuming that this * The symbol Ja" indicates the product ia-ia- 5a . . takenw times as factor; Sa is defined as equal to the identity. 54 ABSTRACT GROUPS (Ch. Ill equivalence remains true when w is any negative integer. In particular, two elements of G which satisfy the equation SaSfi = 1 are said to be the inverses of each others and S0 = Sa~^ is denoted by Sa~^. Elements of order 2 are their own inverses; but all other elements, besides the identity, go in pairs, composed of an element and its inverse. In particular, every possible group contains an even number of elements, which may be zero, of every order which exceeds 2. Since SaSp . . , Sx-Sx~^ . . . Sp-'^S-^ = 1, it results that the iilverse of SaSp . . . s\ is Sx~^ . . . Sp~^Sa~\ and that If all of the elements Sa, Sp, . . . , Sx are of order 2, then (SaSfi . . . Sx)-^=Sx . . . SpSa. In particular, if the product of two elements, Sa, Sp of order 2 is also of order 2 the elements are commutative, that is SaSfi = S0Sa. If the elements Sa,- sj^, . . . , 5^* = 1 do not include all the elements of G, they represent a subgroup of G. By exactly the same arguments as were used to prove that the order of a sub- stitution group is divisible by the order of each of its subgroups (9), it can be proved that the order of an abstract group is divisible by the order of each one of its subgroups. Hence it results that ^ is a divisor of g. 23. The Cyclic Group.* By definition the cyclic group is generated by a single element, and every group which can be generated by a single element is cyclic. If the order of such a group is g, it contains at least one element 5 of the order g. The ordel" of s*, m being an arbitrary positive integer, is g/d, d being the greatest common divisor of m and g. If <l>{g) reprc- Many of the results of this section can be deduced from properties of the n roots of unity. In fact, these n roots form a cyclic group with respect to multi- plication. 23] THE CYCLIC GROUP 55 cnts, as usual, the number of natural numbers prime to g and not greater than g, then G involves exactly <l>(g) elements of order g, and hence it involves <f>(g) generators. All cyclic groups of the same order are simply isomorphic, and hence we shall say there is one and only one Cyclic group whose order is an arbitrary natural number. This group of order g may be represented by the ggth. roots of unity, when they are combined by multipUcation. If m is prime to g, then .5'" generates s; that is, every element is generated by any one of its powers whose index is prime to the order of the element. This theorem is included in the statement that if J, is any number such that the highest common factor of g and k is d, then will s^ generate the cyclic subgroup of order , g/d, which is also generated by s*^. It should be observed that a necessary and sufficient condition that two elements of G generate each other is that they have the same order, and a necessary and sufficient condition that one of these elements generates the other is that the order of the former is divisible by that of the latter. /"/ Let./ be an elemen{ of any group such that f^ is of order n. If all the prime factors of m are also in n. it results from this that the order of / is equal to mn. In general, the given con- dition implies that the order of / is a divisor of mn and a mul- tiple of m'n, where m' is the quotient obtained by dividing m by its largest factor that is prime to n. By means of substi- tutions it is easy to show that / may be so determined that its order is any arbitrary naultiple of w' that divides mn, when- ever the only restriction on / is that the order of /" is n. If k is any divisor of g, so that kh=g, there must be at least one subgroup of order k in the cyclic group G. This contains <t>{k) generators. The theorem that G cannot contain more than one subgroup of order k results immediately from the fact that the dements of such a subgroup must be as follows: since their kih powers are equal to the identity. Hence a cyclic group contains one and only one subgroup whose order is any 50 ABSTRACT GROUPS (Ch. Ill given divisor of the- order of the group. In particular, every subgroup of a cyclic group is cyclic. When g is even, G contains an element of order 2, and every element of G which is the square of another element of G is the square of exactly two such elements. When g is odd, every element of G is the square of only one element of G. In gen- eral, the /;th powers of all the elements of G are the elements of a group C whose order g' is the quotient obtained by divid- ing g by the highest common factor h oi g and k. Each ele- ment of G' is the ^th power of exactly h elements of G. The continued product of all the elements of G is of order 2 or the identity according as g is even or odd. If g is the product of two numbers which are relatively prime, then G is the product of two cyclic subgroups whose orders are these two numbers, and the number of elements of highest order in G is the product of the numbers of the elements of highest orders in these two cyclic subgroups. This is equivalent to the well-known formula that (pirn) = <f){m i) (f)(m2) , whenever mi, nii are relatively prime and mim2 = m. The determination of all the subgroups of G is equivalent to the determination of all the factors of g. If g be written in the form g = P\"p2* ' Pp"''i Pi, p2, , pp being distinct primes, the number of subgroups of G, including the identity, must therefore be (ai + l)(a2 + l) . . . (ap+l)-l. Since each element of a group generates a cyclic group, it is clear that cyclic groups are of fundamental importance. When g is a prime number p, every clement of G besides the identity generates G, and hence this G docs not involve any subgroup besides the identity. The cyclic group of order p is therefore the only possible group of this order. That is, while there is one and only one cyclic group of every possible order, there are orders for which no non-cyclic group exists. These orders include all prime numbers. The smallest com- posite number which is not the order of any non-cyclic group is 15 (cf. Ex. 3, 11). This is a special case of the theorem (which will be proved in 70) that two necessary and sufficient 24] PROPERTIES OF TRANSFORMS 67 conditions that there is only one group of order g are that g is not divisible by the square of a prime number and that none of its prime factors is a divisor of the number obtained by diminishing by unity another such factor. EXERCISES 1. Prove that the 8 natural numbers which are less than 15 and prime to 15 constitute a group with respect to multiplication (mod 15), which is not simply isomorphic with the group formed similarly by the numbers which are less than 24 and prime to 24. 2. The smallest group of multiplication which involves the two matrices 1 0-1 1 1 is of order 8. Is this group simply isomorphic with either of the groups of Exercise 1? Find the six other matrices of this group. 3. To which of the three groups of the preceding exercises is the group of movements of the square simply isomorphic? To which is the group formed by the 8 numbers less than 20 and prime to 20, with resf>ect to multiplication (mod 20), simply isomorphic? 4. Find two groups whose product is the cyclic group of order 36, and determine the number of elements of each order in this group. 5. Including the identity there are five complete sets of conjugate elements in the group of movements of the square. Determine the ele- ments of each of these sets. 6. Do the numbers 2, 4, 6, 8 form a group with resp>ect to multiplica- tion (mod 10)? If so, is this group simply isomorphic with the group formed by 1, 1, V^Tl, V^? Which of the four numbers 2, 4, 6, 8 corresponds to the identity? 7. If 5* is of order 2 find two substitutions of orders 12 and 4 respect- ively which may be used for s. 8. Give the orders of all the pQssible cyclic groups having only two distinct generators. 24. Properties of Transforms. In 9 we considered the transform of a substitution. As the concept of transforming is very useful we shall develop the properties of this operation more fully at this place. Suppose that s-Hs = r. 58 ABSTRACT GROUPS [Ch. Ill By raising each member of this equation to the /3th power there results the equation Hence the theorem: if an element transforms a generator of a cyclic group into its ath power it transforms every element of this cyclic group into its ath power. From the first equation we can also deduce the equation s-H^ = tf^. Since an element and its transform are of the same order, it is necessary that a be prime to the order k of /. Hence If generates t, and from Euler's generalization of Fermat's theorem it results tKat if m is the lowest power of a such that a'" = l(mod k) then w is a divisor of 4>{k). Since 5"* is the lowest power of s which is commutative with /, it follows that m divides the order of s. For instance, an element of order 3 could not transform an element of order 5 into any power of itself, except the first power, since the numbers 2, 3, 4 belong to the exponents 4, 4, 2 respectively modulo 5. If we form the successive transforms 5-%5 = /i, S-^tiS = t2, . . . , S-^tn-iS = tn, it results that for a sufficiently large value of n we have 4 = 4, k<n since the order of S is finite. This implies that /4-a = /*+, a being any positive integer, and if n is the smallest subscript for which this relation is true then ^ = 0. That is, the trans- forms t\, /2, . . . , /o-lj ^ repeat themselves in the given order if the powers of s trans- form to into n distinct elements. Since it results that n must divide the order of s. If this order is a prime number />, n is either 1 or p. That is, if s is not com- 25] CONSTRUCTION OF GROUPS 59 mutative with t it transforms t into n distinct elements where n divides the order of s; whenever this order is a prime number, n is equal to the same prime. Instead of transforming to by the different powers of s we may transform it by all the elements of a group G. If this is done, there results a set of elements, each of which has the same order, and each is transformed into all the others by the elements of G. This set is called a complete set of conjugates of /o under G. In particular, all the elements of G can he separated into distinct complete sets of conjugates as regards G, and this separation can be performed in only one manner. The elements of G which transform to into itself form a subgroup of G, and the number of the elements in the complete set of conjugates to which to belongs is equal to the order of G divided by the order of this subgroup. By transforming all the elements of a group G by the same element / there results a group which is simply isomorphic with G, since we obtain a (1, 1) correspondence by making each element correspond to its transform with respect to /. If t transforms G into itself the resulting simple isomorphism is an automorphism of G. 25. Construction of Groups with Invariant Subgroups. Let Si, 52, . . , Sg = G be any group and suppose that / trans- forms all the elements of G into elements of G (i.e., t trans- forms G into itself), and that T is the lowest power of / which occurs in G. The following rectangle 1, S2, . . ., Sg t, S2t, p-\ S2P-\ ., s,t /7-1 is composed of distinct elements since /' cannot be an element of the form sj^\ where 8i < 5<7, and all the elements of a row are distinct from each other and also from the elements of each of the preceding rows. In order to prove that the elements of this rectangle repre- sent a group, it remains only to show that no additional element 60 ABSTRACT GROUPS [Cii. Ill can be obtained by Combining the elements in every possible manner. This fact results from the equation Hence the theorem : I J t transforms a group G into itself and if /* is the lowest power of t which occurs in G, then t and G generate a group whose order is b times the order of G. This theorem is very useful in the construction of groups. The theorem which has just been proved can be readily extended by replacing the cyclic group generated by / by any group H. It has been observed that all the common elements of G and II constitute a subgroup of both of these groups, viz.', the cross-cut of G and H. By replacing the first column of the given rectangle by elements from the different co-sets of H as regards this cross-cut, we arrive at a more general theorem which may be stated as follows: If all the elements of a group H transform G into itself, tlien H and G generate a group whose order is the order of G multiplied by the index under H of the cross-cut of G and H. It is easy to verify that all the elements of the group generated by G and H transform G into itself. Hence G is an invariant subgroup of this group. It was observed in 10 that when 1, S2, Ss, . . . , Sy is a subgroup of the group G it is always possible to arrange all the elements of G in both of the following ways so that no element is repeated: 1, ^2, S-3, . . ., Sy 1, 52, ^3, . . ., Sy t2, 52^2, 53/2, . . -, Syt2 h, t2S2, t^Si, . . ., t2Sy h) S2t\, 53/x, . . ., Sytx t\, t\S2, IkSs, > ^X^7 In these arrangements the elements of the first column do not necessarily constitute a group. By interchanging rows and columns it becomes evident that such an arrangement is possible when the elements of the first row do not form a group. The question arises whether all the elements of G can be arranged in the given manner even when neither the first row nor the first column is a subgroup of G. That such an arrange. 2Gl THE DIHEDRAL AND THE DICYCLIC GROUPS 61 ment is sometimes possible follows from the fact that when G is the symmetric group of degree 4, we may use for the first row the elements of any two Sylow subgroups of order 8, and for /2 one of the substitutions of order 4 in the remaining Sylow subgroups of order 8. Hence the given rectangular arrangement does not always imply that either the first column or the first row is composed of the elements of a subgroup. 26. The Dihedral and the Dicyclic Groups. Let S\ and 52 represent any two elements of order 2. Their product S\S2 is transformed into its inverse by each of the elements S\ and 52- Each of the elements of the cyclic group generated by S\S2 is therefore transformed into its inverse by each of the two generators S\ and 52. In particular, this cyclic group is trans- formed into itself by each of these generators. Hence two elements of order 2 generate a group whose order is twice the order of the product of these elements. This group is called the dihedral group. When S\S2 is of order 2 the group generated by 5i and 52 is the non-cyclic group of order 4. There is at least one dihedral group of ever>' even order greater than 4, since the group of movements of the regular polygon of n sides is clearly such a group. Moreover, there is only one abstract dihedral group of order 2, > 1. In fact, if there were two such groups their cyclic subgroups of order n could be made simply isomorphic. Since each of the remain- ing elements of both groups would be of order 2 and would transform each element of this cyclic subgroup of order n into its inverse, any two of these remaining elements could also be made to correspond, and thus a simple isomorphism between the two groups could be established. Hence there is one and only one dihedral group of every even order greater than 2. The fact that there is at least one dihedral group of order 2 can also be easily established by means of substitutions. In fact, if n is even we may let S\=a\a2-aiaA- . . . -a^.iO^, 52=0203- . o-20-i- 62 ABSTRACT GROUPS [Ch. Ill If n is odd, these substitutions can be selected as follows: 52 = 02^3- . . . an-3n-2-7i-l<3fn. Since the product of S1S2 in each case is a cyclic substitution of order n it results that 5i and 52 generate the dihedral group of order 2m. The non-cyclic group of order 4 is the only dihedral group which does not involve non-commutative elements. A group which contains no non-commutative elements is called com- mutative or abelian. Since there is one cyclic group of every order and one dihedral group of every even order greater than 2,' there must be at least two groups of every even order greater than 2. When this order exceeds 4 one of these two groups, whose existence has been proved here, is abelian while the other is non-abelian. Instead of defining the dihedral group of order 2 as the group generated by two elements of order 2 whose product is of order , it could also have been defined as the group gene- rated by a cyclic group H of order n and an element of order 2 which transforms every element of H into its inverse. Both of these definitions of the dihedral group are very useful. If n is even we can find an element / of order 4 which transforms every element of // into its inverse and has its square in H. The group of order 2w generated by H and this / is called th(^ dicyclic group whenever >2, and there is one and only om such group of every order which is divisible by 4 and exceeds 4. The smallest dicyclic group is the group of order 8 generated by the four quaternion units 1, i, j, k. This is known as the quaternion group. Its properties were studied by W. R. Ham- ilton. The fact that there is no more than one dicyclic group of a given order can be at once proved by proving that two such groups of the same order are simply isomorphic. The exist- ence of this group for every even value of n may be proved by means of substitution groups as follows: Write a transitive dihedral group of order 2 on two distinct sets of letters and 27) REGULAR SUBSTITUTION GROUPS 63 establish a (1, 1) correspondence between the substitutions. Let t be a substitution of order 2 which is commutative with each substitution of this dihedral group and transforms the two systems of intransitivity into each other. The required dicyclic group of order 2 may now be con- structed by extending the cyclic subgroup of order n in the given dihedral group by means of a substitution of order 4 which is the continued product of /, one of two generating substitu- tions of order 2 of this dihedral group, and the substitution of order 2 which is generated by the cyclic group of order n in one of the two transitive constituents of this group. Hence there exists one and only one dicyclic group of order 4w, where n is any positive integer exceeding unity. Every dicyclic group is non-abelian. We have now estab- lished the existence of three distinct groups of every order which exceeds 4 and is divisible by 4. One of the fundamental prob- lems of abstract group theory is a determination of all the possible groups of a given order. A considerable number of special cases have been solved, but the general solution of this problem seems to he far beyond the present developments of this subject. As early as 1854 Cayley determined the five possible groups of order 8. 27. Representation of a Group as a Regular Substitution Group. Cayley's Theorem. We proceed to prove that every abstract group G of finite order can be represented as a regular substitution group. Let G=\, S2, . . . , Sg and consider the square array of g^ elements formed as follows: 1, ^2, ^3, . . ., Sf 52, S2^, 53^2, . . ., SgS2 53, S2S3, 53^, . . ., SfS3 Sg, S2St, S3Sg, . . .. Sg If we regard the substitutions by means of which each of these lines may be obtained from the first, we obtain a substitution group on g letters, and each substitution besides the identity involves all of these letters. As no two of these substitutions 64 ABSTRACT GROUPS (Ch. Ill are identical, this substitution group is of order g and it is simply isomorphic with G. Since each letter of this group U replaced once and only once by every other letter, the substi- tution group is regular. By combining these facts with those of 14 there results the following theorem: Every group of finite order can he represented as a regular substitution group, and two regular substitution groups which are simply isomorphic are also conjugate* The fact that every abstract group of finite order can be represented as a substitution group enables us to use, in the theory of abstract groups, all the theorems of substitution groups which are confined to group properties; for instance, the theorepis relating to subgroups, quotient groups, sets of conjugates, etc. Many of these properties can, however, be studied to advantage from the standpoint of abstract groups, since we are thus led to fix our attention on the essentials and are not distracted by the notation. In some cases, on the contrary, this notation appears to be the simplest means to establish abstract properties. In fact, we shall see later that linear substitution groups also enable us to prove some impor- tant abstract group properties very readily. From the fact that every group can be represented as a regular substitution group it is very easy to derive a simple proof of Sylow's theorem. This proof is as follows: Let G be any group whose order g is divisible by p" but not by /'""^S and represent G as a regular substitution group. Suppose that p^ is the highest power of p which is less than g, and that g9^p since the case when g is a power of p does not require consideration, and consider all the possible substitutions on the g letters of G which are of degree p^ and of order p. Since G is transitive, it cannot transform any of these substi- * This theorem is fundamental, as it reduces the study of abstract groups uniquely to that of regular substitution groups. The rectangular array by means of which it was proved is often called Cayley's Tabic, and it was used by Cayley in his first article on group theory, Philosophical Magazine, vol. 7 (18.'>4), p. 49. The theorem may be called Cayley's theorem, and it might reason- ably be regarded as third in order of importance, being preceded only by the theorems of Lagrange and Sylow. 27) REGULAR SUBSTITUTION GROUPS 65 tutions into itself. It must therefore transform all of them into complete sets of conjugates under G such that each of these sets is composed of more than one substitution. As the total number of these substitutions is prime to p, according to 14, Ex. 7, at least one of these sets of conjugates involves a number m of substitutions, where m is prime to p. Each of these m substitutions is transformed into itself by a subgroup of G whose order is g/m, where m> 1. Hence G contains a subgroup whose order is divisible by p". If thb subgroup is of order p", our theorem is established. If it is not of this order, we have reduced our problem to that of a smaller group whose order is divisible by p". In case Sylow's theorem were not universally true it would clearly be possible to find a smallest group G for which it would not be satisfied. As the preceding considerations estabUsh the fact that such a smallest group does not exist, they constitute a proof of Sylow's the- orem. EXERCISES 1. If a group involves a subgroup whose order is one-half the order of the group this subgroup is invariant. 2. If the order of a group is pq, p and q being prime numbers and p>q, this group is cyclic unless p1 is divisible by q. In the latter case there are exactly two groups of order pq. 3. Every simple group of composite order can be represented as a non-regular transitive substitution group. Suggestion: Consider the substitutions according to which any com- plete set of conjugate substitutions or subgroups are transformed under the group. 4. There are exactly two abstract groups of order 4.* Suggestion: Represent the possible groups as regular substitution groups. * The non-cyclic group of order 4 is known under various names. AmonR these are the following: Axial grpup, four-group, fours group (Vicrergruppc). quadratic group, anharmonic group, and group of the general rectangle. For the use of these terms, in order, the reader may consult the following: Picrpont, Annals of Mai hematics, vol. 1 (1900), p. 140; Bolza, American Journal of ilatke- - -itics, vol. 13 (1891), p. 75; B6cher, Introduction to Higher Algebra, 1907, S7; Burnside, Theory of Groups of Finite Order, 1912, p. 444; Capclli, IslUu- zioni di analisi algebrica, 1909, p. Ill; Miller, American Maikcmatical itoniUy, vol. 10 (1903), p. 217. 66 ABSTRACT GROUPS [Ch. HI 6. Every group of order p'^, p being any prime number, is abelian, and hence there are exactly two such groups for every prime. Suggestion: If such a group is non-cyclic it contains p-\-\ subgroups of order p. These could not constitute a complete set of conjugates. 6. If the order of a group is the double of an odd number, the group contains an invariant subgroup of half its own order. Suggestion: Write the group as a regular group and observe that it contains negative substitutions. 28. Invariant Subgroups and Quotient Groups. In 13 we gave examples of invariant subgroups and of quotient groups as related to intransitive substitution groups. The term invariant subgroup was defined in 9. Another but equivalent definition is based on the following considerations: If H is any subgroup of G, then G can be represented in either of the following two forms: G^H-\-Hs2+ . . . +Hsx ^H+S2-'H^ . . . ^s^-'H. The co-sets HSa, a = 2, . . . , X, are called right co-sets, while those of the form Sa~^H are called left co-sets. A necessary and sufficient condition that H is an invariant subgroup of G is that every right co-set of G as regards H is equal to some left co-set of G as regards H. If this condition is satisfied the totality of the left co-sets is identical with the totaUty of the right co-sets. This theorem may be stated more generally as follows: A necessary and sufficient condition that H is invariant under the substitutions of the right co-set HSa is that Hsa^SaH. When // is an invariant subgroup of G, the augmented co-sets of G as regards H may therefore be regarded as elements of a new group Q, H being the identity of Q* This group Q is the quotient group G/H of G as regards H (cf. 13). There is an {h, 1) isomorphism between G and Q, h being the order of H. When // is the identity this isomorphism reduces to a simple isomorphism, and G and Q are the same * E. Galois first directed attention to the invariant subgroups and thii: important properties. The fact that each invariant subgroup gives rise to a quo- tient group is fundamental. 28] INVARIANT SUBGROUPS 67 abstract group. Whenever //> 1 the isomorphism is said to be multiple. The groups H and Q are called complementary groups as regards G, and the product of their orders is equal to the order of G. The fact that two different groups may have the same complementary groups results directly from the dihedral and the dicyclic groups. Let q be any element of Q and let s be any one of the elements of the corresponding co-set. The order of s must be divisible by the order m of q, since s^ is the lowest power of s that occurs in H. If m is a power of a prime p then there is an element in the corresponding co-set whose order is also a power of p, since the group generated by H and this co-set must involve a larger subgroup whose order is a power of p than // does. Hence the theorem: The order of any element of a quotient group divides the orders of all the elements of the corresponding co-set, and if this order is a power of a prime number the given co-set involves an element whose order is a power of the same prime. As a special case of this theorem it may be observed that every invariant subgroup of index 2 under any group includes all the elements of odd order contained in this group. Two elements which belong to the same co-set as regards an invariant subgroup are sometimes called equivalent with respect to this invariant subgroup. They are also said to be congruent with respect to this invariant subgroup as a modulus. It should be observed that an invariant subgroup has many of the prop- erties of a modulus in elementary number theory. To a smaller extent these properties belong to all subgroups, and the terms equivalent and congruent are sometimes used in connection with any subgroup. From the separation of the elements of a group into co-sets it results directly that every subgroup of index p under any group includes a p'th, p < p, part of the elements of every other subgroup of G. We proceed to prove that p<p whenever the two distinct subgroups Gi, G2 in question are conjugate under G. Suppose that p = p. It must therefore be possible to write all the elements of G in the form S1S2, where si 13 any element of Gi and S2 is any element of G2. Hence all the con- 68 ABSTRACT GROUPS (Ch. Ill jugatcs of Gi under'G are also conjugates under G2. This is, however, impossible, when Hi and ^2 are conjugate since the elements of G2 cannot transform d into C2. As the assump- tion that p' = p has led to an absurdity, it has been proved that the index of the cross-cut of any two distinct conjugate subgroups under one of these subgroups is always less than tJie index of iJtese subgroups under the entire group. To find a simple illustration of this fundamental theorem, supp>ose that the index of Gi under G is 2. It follows then directly from this theorem that if Gi and G2 are two conju- gate subgroups, the cross-cut of Gi and G2 must have an index wjiich is less than 2 under each of these subgroups. Hence this index is 1, and Gi, G2 are identical. Hence the given theorem includes as a special case the theorem that a sub- group of index 2 under any group is invariant. This fact can also be readily proved in other ways. Cf. preceding Exercises. If the invariant subgroup U is composed of all the invariant elements of G it is called the central of G and its complementary group is known as the central quotient group of G. When this central quotient group is abelian G is said to be metabelian* The central quotient group is also called the group of con- gredient isomorphisms of G. It is clear that the central of G is always abelian. For instance, 1,-1 constitute the central of the quaternion group, and the central of an abelian group coincides with the group. In a non-abelian group the order of the central cannot exceed the order of the group divided by 4, and if this is the order of the central of G, the central quotient group is the axial group. It is easy to prove that tfte central quotient group is always non-cyclic. 29. Commutators, Commutator Subgroup, uid the <^sub- group. The element or operator f s'H-^st is called the co7n- mutator of s and /, while its inverse is the commutator of / and s. When 5 and / are commutative their commutator is * W. B. File, Proceedings of the American Association for the Advancement of Science vol. 49 (1901), p. 41. t The elements of a group are also called operators or operations. We shall hereafter use the terms element, operator, and operation interchangeably, since all of these terms are commonly found in the modern literature of group theory. 29] COMMUTATOR SUBGROUP 69 the identity and vice versa. By writing the commutator of 5 and / in the form s-H-^st = c, or t-^st = sc, it is clear that the commutator represents an operator which must be multiplied into another operator to obtain its conjugate. A group G of order g has g^ commutators, but no more than g of them can be distinct. In general, these g^ commutators generate a sub- group of G, known as the commutator subgroup* of G, or the first derived group of G. When G coincides with its commutator subgroup it is said to be a perfect group. To prove that the commutator subgroup of G is invariant under G it is only necessary to prove that the transform of a commutator of G as regards any element of G is also a com- mutator of G. This fact results immediately from the equa- tion r-^s-H-^str = r-h-^r-r-H-^r-r-^sr-r-Hr = si-Hi-^Siti. It should be emphasized that a group may have many invari- ant subgroups, but it has only one commutator subgroup. The quotient group which corresponds to the commutator subgroup is known as the commutator quotient group. This quotient group is always abelian, since the commutator of two of its elements must correspond to the commutator of any two of the corresponding elements of G and hence it must be the identity. As the commutator quotient group is abelian, and as every invariant subgroup which is complementary to an abelian quotient group must include the commutator subgroup, it results that this subgroup is the smallest invariant subgroup that is complementary to an abelian quotient group. In fact, every invariant subgroup which is complementary to an abelian quotient group includes the commutator subgroup. From the given definition of a perfect group it results also that every simple group of composite order is a perfect group, but the con- * The term commutator subgroup is due to R. Dedekind, but the fundamental properties of these subgroujjs were first published by G. A. Miller, Quarlerly Journal of Malhematks, vol. 2S (1S9C.), p. 20<). Their usefulness was quickly recognized, and they have entered largely into the recent group theory litcralure. 70 ABSTRACT GROUPS [Ch. Ill verse is not necessarily true. That is, there are perfect groups which are not also simple, as we shall see later. If the elements of a commutator belong to two invariant subgroups of a group G, this commutator is contained in the cross-cut of these invariant subgroups. Hence it results that if two invariant subgroups of G have only the identity in com- mon, every element of each one of these subgroups is commutative with every element of the other. If the elements of the commutator s'^t-'^st are permuted in every possible manner, there result eight operators which may be distinct and may all differ from the identity. These eight operators are: s-'^t-'^st, t'^sts-^, sts'^t-^, ts-^t-^s, t-^S'Hs, st-^s~H, tst-^s-\ s'Hst-^. All of them can be obtained from any one of them by means of the substitution group of order 8 on the four factors. It is evident that each of these 8 commu- tators has the same order. To prove this it may first be observed that the order of any product of n operators is invariant as regards the cyclic group oj permutations of these factors, since such permutations are equiv- alent to transforming by elements. If reversing the order of these n factors does not affect the order of the product, this order is invariant as regards the dihedral group of permutation of its n factors. In particular, the order of the product of n ele- ments of order 2 is always invariant as regards the dihedral group of permutations of the n elements. Reversing the order of the factors of a commutator cannot affect the order of this commu- tator, since it is equivalent to a cyclic permutation of the fac- tors of its inverse. The given eight commutators, involving s, t and their in- verses, are contained in the commutator subgroup of the group generated by 5 and /, but they do not necessarily generate this subgroup. Since four of them are the inverses of the other four, it is clear that no more than four of them are distinct when their common order is 2. The most general definition of a commutator is, "the product of the transform of an element and its inverse." Whenever an element can be written as such a product, it may be called a commutator. When we speak 29] COMMUTATOR SUBGROUP 71 of the commutator of a group it is assumed that the elements of the commutator are elements of the group in question, and hence it may happen that only a small number of the elements of the group are commutators. For instance, the identity is the only commutator in an abelian group. We shall see later that every possible group element may be regarded as a com- mutator of some two elements. When s and / are both of order 2 their commutator is the square of their product and is trans- formed into its inverse by both of its elements. The importance of the concept of commutator is largely due to the fact that the commutators of a group represent those operators which must be multiplied on the right into a given one of a complete set of conjugate operators to obtain all the others. Hence the order of the commutator subgroup of a group cannot be less than the number of conjugate operators in its largest complete set of conjugates. Since the commu- tator subgroup is unique, it must evidently correspond to itself in every possible automorphism of the group. A subgroup which has this property is said to be a characteristic subgroup * Hence it results that if a group is not perfect, its commutator subgroup is a characteristic subgroup. It was observed in 3 that any set of operators belonging to any group G of finite order is called a set of independent generators of G provided that these operators generate G and that none of them is contained in the group generated by the rest of them. All the operators of G can be divided into two categories, having no common operator, by putting into one category all those which occur in at least one of the possible sets of independent generators of G, and into the other cate- gory those which do not have this property. The operators of the second category constitute a characteristic subgroup of G, which has been called by G. Frattini the (f>-subgroup of G.f If H is any maximal subgroup of G it is evidently always possible to select at least one set of independent generators * G. Frobenius, Berliner Sitzungsberkhie, 1895, p. 183. t G. Frattini, AUi delta Reale Accademia dei Lincei, Rendiconti, scr. 4, vol. 1 (1885), p. 281. 72 ABSTRACT GROUPS [Ch. Ill of G in such a manner that it includes any arbitrary one of the op)erators of G which are not contained in H, while the remain- ing operators of the set belong to //. Moreover, there is at least one maximal subgroup of G which does not include any given one of the independent generators of a particular set of independent generators of G. Hence it results that the 0-sub group of G is the cross-cut of all the maximal subgroups of G. This useful second definition of the (^-subgroup is also due to Frattini. If a 0-subgroup of the group G involves a non-invariant subgroup or a non-invariant operator, this subgroup or operator cannot be transformed into all its conjugates under G by the operators of the (^-subgroup. That is, every complete set of conjugates of the </)-subgroup is an incomplete set of conju- gates under G whenever the former set involves more than one element. If this were not the case all the operators of G which would transform one of these elements into itself would form a subgroup which would not involve all the operators of the (^-subgroup of G. This subgroup could not be maximal, since it does not involve the 0-subgroup. As any maximal subgroup obtained by extending this subgroup by means of operators of G could also not involve the (^-subgroup, we have proved the theorem: If the <t>-subgroup of a group G involves a non- invariant operator or subgroup, the number of conjugates utider G oj this operator or subgroup is greater than the number of the corresponding conjugates under the <t)-sub group. An important special case of this theorem was noted by Frattini, who observed that the </)-subgroup of any group involves only one Sylow subgroup for every prime which divides the ordci of the (^-subgroup. In other words, every 0-subgroup is the direct product of its Sylow subgroups, and hence we can alwa} s reach the identity by forming successive 0-subgroups, starting with any given group. If a group can be represented as a non- regular primitive substitution group of degree , its w subgroui)^ of degree 1, each being composed of all its substitution- which omit a letter, are maximal and have only the identity in common. Hence it results that the 4>-subgroup of every primitive substitution group is the identity. 30] SIMPLY ISOMOUPHIC GROUPS 73 EXERCISES 1. If an intransitive group of degree h contains exactly it transitive constituents, the average number of letters in all its substitutions is nk. 2. Prove that the group of the square involves an invariant sub- group leading to the four-group as a quotient group and that this invariant subgroup is its commutator subgroup. 3. All the elements which are common to all the subgroups of a com- plete set of conjugate subgroups constitute an invariant subgroup. 4. To every invariant subgroup of a quotient group there corresponds an invariant subgroup of the group, and to every subgroup which involves a given invariant subgroup there corresponds a subgroup in the quotient group corresponding to this invariant subgroup. 5. If every element of a group is raised to the same power and if this power is prime to the order of the group, each element of the group is found once and only once among these powers. 6. The commutator subgroup of the symmetric group of degree n is the alternating group of this degree, and the alternating group of degree n is perfect whenever w>4. 7. A necessary and sufficient condition that the ^-subgroup of a cyclic group is the identity is that the order of this group is not divisible by the square of a prime number. 30. Simply Isomorphic Groups. One of the most miportant and most difficult problems in group theory is to determine whether twD given groups of the same order are simply isomor- phic or not. If they are simply isomorphic they are identical as abstract groups and vice versa. Two cyclic groups of the same order are always simply isomorphic and a cyclic group cannot be simply isomorphic with a non-cyclic group. One of the most useful theorems as regards simply isomorphic groups may be stated as follows: Two groups of the same order Gi, G2 are simply isomorphic if they contain two simply isomorphic invariant subgroups Hi, H2 respectively, and are generated by tJiese subgroups and two elements /i, /2 such that if tx" is the lowest power of t\ which occurs in Hi , then /2 is tlie lowest power of tz that occurs in H2, afid /i", to" correspond in the given simple isomorphism of Hi, H2. Moreover, it is assumed tliat t\, /2 transform corre- sponding generators of Hi, H2 into corresponding elements in the given simple isomorphism. To illustrate the use of this theorem, let Gi, G2 represent two 74 ABSTRACT GROUPS [Ch. Ill dihedral groups of order 2 and let II i, H2 represent their cyclic subgroups of order n. Let tu h represent any two elements of Gi, G2 respectively but not contained in Hi, Il2- The common order of /i, /2 is 2, and /i, /2 transform corresponding generators of III, II2 respectively into corresponding elements, since they transform all the elements of these subgroups into their inverses. Hence this theorem includes the known theorem that two dihedral groups of the same order are always simply isomorphic. The given illustrative example of the use of the theorem in question may also serve to point out the way toward a proof. In fact, if H\, H2 are arranged in a simple isomorphism and if the products obtained by multiplying corresponding opera- tors by h^, /2^ respectively, /3 = 1, 2, . . . , a\, are placed in correspondence, we obtain a simple isomorphism between Gi, G2. In fact, /i", t2^ transform all the corresponding opera- tors of H\, B.2 into corresponding operators because they trans- form generators of B.\, B.2 in this manner. It may be observed that this theorem may be employed to prove that two cyclic groups of the same composite order are simply isomorphic if it is assumed that two cyclic groups of the same prime order have this property. It results immediately from the given theorem that if two abelian groups of the same order involve only operators of the same prime order besides the identity, they must be simp!} isomorphic. Among the most important simple isomorphisms are those in which the operators of the same group G are placed into a (1, 1) correspondence in such a way that an automorph- ism of G is obtained. We have seen ( 24) that any opera- tor which transforms G into itself effects an automorphism on its elements. Moreover, any automorphism of G can always be brought about by transforming G by some operator which transforms G into itself. To prove this statement we shall employ a method which has been illustrated in 14 but which we desire to exhibit more fully. Represent G as a regular substitution group and establish an arbitrary automorphism of G. We may suppose that all the substitutions begin with the same letter, so that the second 301 SIMPLY ISOMORPHIC GROUPS 76 letters of the corresponding substitutions exhibit a substitu- tion by means of which we can transform one of these groups in such a way that the first two letters of all the corresponding substitutions are identical. After this has been done all the corresponding letters of the corresponding substitutions must be identical, so that the given substitution may be used to effect the given automorphism. This follows immediately from the fact that the two regular groups in which all of the correspond- ing substitutions have the first two letters in common are still simply isomorphic, and if in a given substitution c is replaced by d, this substitution is the product of the inverse of the sub- stitution in which a is replaced by c and the substitution in which a is replaced by d. Hence in the corresponding subsU- tution of this automorphism c must also be replaced by d, since a regular group contains only one substitution in which a given letter is replaced by another given letter. In view of the importance of this theorem we shall give an iUustrative example. Consider the following automorphism of the symmetric group of order 6: 1 1 abc'def acb-dfe ach ' dfe abc dcf ad-bf-ce adbf'ce ae-bd-cf af-be-cd af-be-cd ae-bd-cf The substitution which transforms the former of these two groups so that the first two letters in each pair of corresponding substitutions are identical is bc-ef This substitution also transforms the former of these corresponding groups into the latter. The theorem that every automorphism of G may be obtained by transforming G by operators which transform G into itself is not only useful in finding the possible automorphisms, but it may also be used to determine the totality of the sub- stitutions which transform a given group into itself. 76 ABSTRACT GROUPS [Ch. Ill EXERCISES 1. The substitutions which represent the transformations of the sym- metric group of order 6 into all its possible automorphisms constitutes a group which is simply isomorphic with this symmetric group. 2. If a substitution of order 2 transforms d into d it must ako trans- form Gi into Gi. 3. Any group G of order g that involves an invariant operator s of order h can be extended by means of an operator / of order nh which is commtitative with every operator of G and satisfies the equation t'*=s so as to obtain a group of order gn. Suggestion: Write G as a regular group on n distinct sets of letters and establish a simple isomorphism between these groups. Let /i be a substitution of order n which permutes the corresponding letters of this intransitive group and is commutative with each of its substitutions. For / we may use the product of h and the substitution s in one of the regular constituents. 31. Group of Inner Isomorphisms. If all the elements of a non-abelian group G are transformed by any one of its own elements, the elements of G are permuted according to a certain substitution. By transforming the elements of G successively by all the elements of G there result a series of substitutions which constitute a group known as the group of inner isomorph- isms oj G. This group is also called the group of cogredient isomorphisms of G, and it is simply isomorphic with the central quotient group of G ( 28). A necessary and sufficient condi- tion that G is simply isomorphic with its group of inner iso- morphisms is that the central of G is the identity. If G admits isomorphisms which cannot be obtained by transforming all the elements of G by its own elements, they are called outer, or contragredient, isomorphisms. In this case the group of inner isomorphisms is clearly an invariant sub- group of the group of isomorphisms of G. One of the most useful properties of the group of inner iso- morphisms I\ of G is that I\ contains tJie same number of Sylow subgroups of every order as G does, provided we call the identity a Sylow subgroup of order /> whenever the order of the group 1 1 is not divisible by p. The truth of this fact becomes evident if we observe that a Sylow subgroup of /i has exactly the same number of conjugates under /i as the corresponding Sylow 32) FROBENIUS'S THEOREM- 77 subgroup of G has under G. In particular, an abeUan group contains only one Sylow subgroup of every order. - In 13 we defined the term direct product as regards sub- stitution groups. In general, a group is said to be the direct product of two subgroups which generate it, provided these two subgroups have only the identity in common and every element of the one is commutative with every element of the other For instance, tfie holomorph of a complete group is the direct product of tJte group and its coyijoint ( 19). This holomorj^h IS also said to be the square of this complete group. A simpler lUustration of a direct product is furnished by the axial group which IS the direct product of any two of its subgroups of order 2. o f As the group of inner isomorphisms of G cannot be cycUc It cannot have a smaller order than 4. If it is of order 4 G contains exactly three abelian subgroups of half its own order- and vice versa. When G is non-abeUan and contains more than one abelian subgroup of half its order, then G has the four-group for its group of inner isomorphisms. As instances of groups which have the four-group for their group of inner isomorphisms we may cite the octic group and the quaternion group. 32. Frobenius's Theorem. In an article published in the Berliner Sitzungsberichte, 1895, page 984, Professor G. Fro- benius developed a very fundamental theorem" which may be stated as follows: // n is any factor of the order g of a group G, the number of the operators in G, ifuluding the identity, whose orders divide n, is a multiple of n. This theorem is evidently true when g is a prime number, and also when n=g. Hence, in a proof by complete induction,' we may assume that the theorem holds for all the groups whose orders involve a fewer number of factors than g does, and also for all factors of g which are larger than n. If we can prove that it remains true for G and n provided these assumptions are made, then the theorem will follow by complete induction for an arbitrary group G and any divisor n of its order. 78 ABSTRACT GROUPS [Cb. Ill The theorem is clearly true for every cyclic group, since such a group has one and only one subgroup whose order is any divisor of the group, and this subgroup includes all the opera- tors of this cyclic group whose orders divide this divisor. To simplify the general considerations which follow, we shall first prove that the theorem applies also to the non-cyclic group of order pq, p and q being distinct primes and p>q. Let iV, represent the number of operators of G whose orders divide x, while Nx represents the number of those operators of G whose orders do not divide x. Hence, in the special case when G is the non-cyclic group of order pq. As Nm is divisible by p we can prove that Np is divisible hyp by proving that Np is divisible by p. The fact that Np is divis- ible by p results directly from the fact that all of these opera- tors are of order q, since G is non-cyclic, and they must occur in complete sets of conjugates under a subgroup of order p, such that each set involves p of these conjugates. We shall now prove, by means of the two given assumptions, the general theorem stated above. If p represents any prime divisor of g/n, G involves, by h>'pothesis, a multiple of np operators whose orders divide np. As Nnp = Nn + N'n, where N'n represents the number of operators of G whose orders divide np but do jiot divide n, it is evident that we have only to prove that N'n is divisible by n in order to prove our theorem. _ The totality of these iV' operators will be represented by A. Suppose that n = p^-h where s is prime to />. It is clear that A is composed of operators whose orders are divisible by p^, and hence it is very easy to see that N'n is divisible by p^-^. In fact, every cyclic group whose order is divisible by p^ must have a multiple of p^'^ distinct generators, since ^(^x)=^x-i(^_l)^ ^(^fn) being the totient of m. Hence it remains only to prove that N'n is also divisible by s. 32] FROBENIUS'S THEOREM 79 Among the operators of A there may be several which have the same constituent P of order p^. All such operators are the direct product of P and operators whose orders are divisors of 5, and all the operators of A may be divided into distinct sets such that each set is composed of all the operators of A which have the same constituent of order p^. We proceed to prove that the number of operators in the combined sets which involve all the conjugates of P under G is divisible by 5, and hence that the total number of operators in A is divisible by 5. To prove this fact, we consider all the operators of G which are commutative with P. These form a subgroup H of order />V, and the quotient group of H with respect to the cyclic group generated by P is of order r. The orders of the operators of this quotient group which divide 5 must also divide the highest common factor (/) of j and r. As the order of this quotient group involves fewer factors than G does we may assume that the number of its operators whose orders divide / is kl. Hence A contains exactly kt operators which have the same constit- uent P. The combined sets which involve no operator of order />^, except the conjugates of P under G, must therefore involve gkt/{p^r) distinct operators, since P has g/ip^r) conjugates under G. Since g is divisible by s and r, it follows that gl is divisible by r^ t being the highest common factor of r and s. Hence 5 is a divisor of gkt/r. As 5 and p^ are relatively prime, 5 must also be a divisor of ght/{p\), and the theorem in ques- tion has been proved. While the number of the operators of G whose orders divide any divisor of ^ is always a multiple of n. it does not follow that groups exist in which the number of these operators is an arbitrary multiple of n. For instance, if p is the highest power of the prime p which divides g, G contains at least one subgroup of order p", according to Sylow's theorem. If G contains only one such subgroup this must be invariant and hence G involves only p"* operators whose orders divide p". If G contains more than one subgroup of order /)". it must contain at least p-\-\ such subgroups, since one such subgroup so . ABSTRACT GROUPS (Ch. Ill must transform any other into at least p distinct subgroups. If G contains exactly /> + l such subgroups they must have in common p"'^ operators and hence they involve exactly (/>+i)(r-r"')+r"'=r+' different operators. Hence G contains exactly p operators whose orders divide P" whenever it contains only one subgroup of order />", and it contains exactly p"^^ operators whose orders divide />* whenever it contains exactly p-\-\ such subgroups. We proceed to prove that when G involves more than /> + l subgroups of order p", it must also involve more than />"+* operators whose orders divide p". To prove this fact we may assume that p^ is the largest number of operators in common to two subgroups {Hi, H2) of order p". Then Hi will transform H2 into p"~^ distinct subgroups. At least p-\-l of these have the same subgroup of order p^ in common, according to the evident theorem that every non-invariant subgroup of a group of order p" is trans- formed into itself by at least one of its conjugates. If /3<a 1, these p-\-l subgroups involve more than />"+^ operators whose orders divide p. If /3=a 1 they must involve exactly p"'^^ operators whose orders divide p". If these /> + l subgroups are such that they are not transformed among themselves by all their operators, there must be another subgroup of order P" in G which involves the />"~^ operators common to the given p-\-l subgroups, and hence G would involve more than />"+' Of)erators whose orders divide p. It remains only to consider the case when each of the given p-\-l subgroups transforms the remaining p subgroups among themselves, and when G contains at least one more subgroup of order />". In this case, this additional subgroup of order p" would contain operators which would not transform the given ^+1 subgroups among themselves, and hence G must contain more than p" '^^ operators whose orders divide p whenever it con- tains more than p-\-l subgroups of order />*. This proves that whenever the number of operators in G whose orders divide p" exceeds p it must be at least />'*'"*, and A 331 ABSTRACT GROUP REPRESENTED BY SUBSTITUTIONS 81 hence it establishes the fact that it is impossible to construct a group G such that its order g is divisible by p, but not by p"'^^ and such that the number of its operators whose orders divide p" lies between />* and p"'^^. EXERCISES 1. If the number of the operators of a group G, whose orders divide an arbitrary divisor d of the order of G, is exactly d, then G must be cyclic. 2. There are no groups with the property that every cyclic subgroup besides the identity is transformed into itself by only its own operators. Cf. Dyck, Mathematische Annalen, vol. 22 (1883), p. 101. 3. Let Xi and n represent two arbitrary numbers, and denote by S\ and ^2 the operations of subtracting n from xi and dividing Xi by n respect- ively. If n is replaced successively by all the numbers resulting from these operations, then Si and ^2 will, tn general, generate the symmetric group of order 6 when Xi^=X2. When Xi^=2X'>, and when Xi*=3x2, these operations will generate the octic group and the dihedral group of order 12 resp)ectively.* 33. Representation of an Abstract Group as a Transitive Substitution Group. In 27 it was observed that every group of finite order can be represented in one and only one way as a regular substitution group. It is often very useful to represent a given abstract group as a transitive substitution group on the smallest possible number of letters. Many of the abstract prop- erties of a group can often be most readily determined if the group is written in this form. Hence we proceed to consider the general question of representing an abstract group G as a non-regular transitive substitution group of degree n. In 12 it was observed that when G is thus represented, it involves n conjugate subgroups Gi, G2, . . . , Gn each of which is composed of all the substitutions of G which omit a given letter. If the subgroup of Gi, composed of all its substitutions which omit a given letter, is of degree na, these n conjugate subgroups are identical in sets such that each set involves a of them. As G is non-regular, there must be at least two such sets, and hence we see directly that G cannot be represented * These groups, together with the four group, have been tailed groups of sub- traction and division, Quarterly Journal oj Mathematics, vol. 37 (1906), p. 80l ^ 82 ABSTRACT GROUPS [Ch. Ill as a non-regular transitive substitution group unless it contains a non-invariant subgroup H which involves no invariant sub- group of G besides the identity. This condition is sufficient as well as necessary in order that G can be represented as a non-regular transitive substi- tution group. In fact, if the elements of G are arranged in a rectangle, where those of such a subgroup H appear as the first row, as follows: 5l, S2, . . ., Sn Sit2, 52/2, . . ., Snt2 Sltn, S2tn, , Sf^tn the lines are permuted as units if all the elements are multiplied on the right by any element of G, since these lines are the co- sets of G as regards H. Hence each element of G may be denoted by the substitu- tion according to which it permutes these co-sets when it is used as a multiplier in the given manner. No two elements of G could permute these co-sets according to the same substitution, since H is non-invariant under G and does not involve any invariant subgroup, besides the identity, of G. This proves the following theorem: A necessary and sufficient condition that an abstract group G of order g can be represented as a transitive substitution group of degree n is that G contains a non-invariant subgroup of order g/n which does not include any invariant sub- group of G besides the identity. In the given method of representing G as a transitive sub- stitution group of degree n it is clear that H corresponds to the subgroup composed of all the substitutions which omit a given letter of this transitive group. All the subgroups of G which correspond to // in one of the possible automorphisms of G give rise to the same transitive substitution group, but no other subgroup can have this property. Hence G gives rise to as many different transitive groups of degree n as it has sets of subgroups of order g/n such that each set includes all those subgroups which correspond in some one of the possible auto- 33] ABSTRACT GROUP REPRESENTED BY SUBSTITUTIONS 83 morphisms of G, and such that these subgroups do not include any invariant subgroup of G besides the identity.* For instance, the symmetric group of order 24 contains three conjugate cyclic subgroups of order 4 and also three conjugate noncyclic subgroups of this order. It contains no other non- invariant subgroup of order 4. Hence this symmetric group appears twice among the transitive substitution groups of degree 6. As the alternating group of degree 4 contains two sets of conjugate subgroups, of orders 2 and 3 respectively, this group can be represented transitively in only two ways besides the regular form. That is, it appears as a transitive group of degree 4 and also as a transitive group of degree 6. It should be observed that these considerations establish another ver>- close contact between the theory of abstract groups and that of transitive substitution groups. It is now easy to prove the theorem, to which we referred in 10, that the co-set multipliers may be so selected that they are the same on the right as on the left. When the subgroup // is invariant this requires no proof. When H does not involve any invariant subgroup of G besides the identity, and G is represented as a transitive substitution group of degree n with respect to H, it may be supposed that H is composed of all the substitutions of G which omit a given letter a. The right co-sets will then be composed separately of all the substitutions of G in which a is replaced by a given letter. In the left co-sets a is replaced by all the letters of a transitive constituent of H if H is intransitive on 1 letters. If H is transitive on n 1 letters the theorem is ev^dent^ Hence it remains only to consider the case when H is intran- sitive. In this case it is evidently possible to select a certain number of left co-sets involving all the substitutions of the same number of right co-sets, and in these the multipliers may be made the same in the left co-sets as in the corresponding right co-sets. Hence the theorem is established in case H con- tains no invariant subgroup of G besides the identity. If // involves such an invariant subgroup but is not itself Bulletin of the American Mathematical Society, vol. 3 (1897), p. 215. 84 ABSTRACT GROUPS [Ch. Ill invariant under G, H will correspond to a subgroup in a quotient group, such that this quotient group can be represented trans- itively with respect to this subgroup. Hence the theorem is also true in this case. For an abstract proof of this theorem the reader may consult H. W. Chapman, Messenger of Mathe- matics, vol. 42 (1913), page 132. EXERCISES 1. If a dicyclic group is represented as a transitive substitution group it must be regular. 2. A dihedral group of order 2m, n>2, can be represented in two and in only two ways as a transitive substitution group. 3. Only one of the five possible groups of order 8 can be represented as a non-regular transitive substitution group. 4. If a simple group of composite order is represented as a transitive group of lowest possible degree it must be primitive. 5. There are five and only five abstract groups of order 12. 34. Historical Note.* The concept of group is one of the oldest mathematical concepts. Even in the development of elementary geometry by the Greeks this concept played a fundamental role, as was pointed out by H. Poincare in an article entitled " On the foundations of Geometry," Monisl, volume 9 (1898), pages 1-43. It was, however, not developed into an extensive theory until a comparatively recent period. In the latter half of the eighteenth century various writers, especially J. L. Lagrange and A. T. Vandermonde, began to lay stress, in their algebraic investigations, upon the elements of the theory of substitutions. On the other hand, L. Euler brought some properties of abelian groups into prominence, especially by his work on power residues. Towards the close of the eighteenth and at the beginning of the nineteenth cen- tury, two Italian mathematicians, P. Rufl5ni and P. Abbati, entered more directly on the study of substitution groups by proving that there are no three or four-valued rational functions of more than four variables, and that a two-valued function must be alternating. During the first half of the nineteenth century a number of other investigators entered this field. Foremost among Cf. Bibliolheca Malhemalica, vol. 10 (1910), p. 317. 34] HISTORICAL NOTE ON GROUPS 86 these were C. F. Gauss, A. L. Cauchy, N. H. Abel, and E. Galois. The first of these helped to lay a foundation for abelian groups by his investigations in number theory, while the other three contributed directly towards the development of the general theory of substitution groups. Abel and Galois solved two fundamental problems in the theory of equations by means of substitution groups, and thus they directed attention to the usefulness of this subject (Cf. Part III). On the other hand, Cauchy ordered and extended the results obtained by his predecessors and contemporaries, and laid a broad foundation for the general theory of substitution groups. Hence Cauchy is frequently called the founder of this theory. The theory of abstract groups grew gradually out of that of substitution groups. In fact, we find even in Cauchy's later writings a tendency to state some results independently of the notation. Cayley's table and Cayley's theorem are very fundamental in this theory, and hence A. Cayley is sometimes called the founder of the abstract theory of groups. In 1856 W. R. Hamilton gave abstract definitions of the groups which are simply isomorphic with the groups of movements of the five ancient regular solids. By these definitions, and the fact that his quaternion units and their negatives form an impor- tant non-abelian group of order 8, he contributed considerably towards an interest in this theory. A solid foundation for an abstract theory implies, however, clear abstract definitions of the terms used. Among the earlier writers one fails to find such definitions. According to E. V. Huntington, Transactions of the American Mathematical Society, volume 6 (1905), page 181, the earliest explicit set of postulates for abstract groups were given by L. Kronccker in 1870. Even at the present time the term group is sometimes used with different meanings as a mathematical technical term.* The earliest separate text-book devoted to group theory is Jordan's Traite des substitutions et des equations algibri- ques, Paris, 1870. This work has become a classic. It was written, however, before the abstract theory was well estab- * Cf. Encycloptdie des Sciences Malhimatiques, tome 1, voL 1, p. 576; vol. 2, p. 243. 86 ABSTRACT GROUPS (Ch. Ill lished. Even Sylow's theorem appeared two years later, and a number of other more recent theorems make it possible to present many subjects in a simpler manner than could be done at the time when Jordan wrote this Traite. The only other separate text-books on groups of finite orders which appeared before the present century are Nctto's SubstitutionentJieorie, 1882, Burnside's Theory of Groups of Finite Order, 1897, and Bianchi's Lezioni sulla theoria dei gruppi di sostituzioni, 1897 (lithographed). Netto's book was translated into Italian by G. Battagb'ni in 1885, and into English by F. N. Cole in 1892. In the older writings on groups, and in some of the more recent ones, a set of distinct elements considered with respect to only one formal law of combination is said to form a group whenever every element obtained by combining an element with itself or with any other element of the set is again in the set. In most cases it is, however, tacitly assumed that these elements obey the other laws which were imposed in our definition of a group. If this is not assumed the set may be said to have " the group property." Cf. M. Bocher, Introduction to Higher Algebra, 1907, page 82. Since the beginning of the present century a considerable number of separate treatises on groups of finite order have appeared. The literature on this subject has been made more accessible by the publication of two somewhat extensive bib- liographies, viz., The Constructive Development of Group-Theory by B. S. Easton, University of Pennsylvania, 1902, and the " Essai d'une bibliographic sur la theorie dcs groupes " by C. Alasia, published in the Rivista di fisica, matltematica e scicnze naturali, Pavia, 1908-10. The main theorems of the theory of finite groups, together with numerous historical data relating to their development may be found in the Encyclopedie des Sciences Mathematiques, tome 1, volume 1, page 532, and in the second edition of Pas- cal's Repertorium der hdheren Mathematik, volume 1, page 168. A list of remarks on the bearing of the theory of groups, exhibiting the wide application of this subject, was published in volume 6 (1914), of the Tohoku Mathematical Journal. CHAPTER IV ABELIAN GROUPS * 35. Invariants. A group is said to be abelian when each of its elements is commutative with every element of the group, 26. Since all the Sylow subgroups of the same order are conjugate under every group, it results that an abelian group can have only one Sylow subgroup of a given order, and hence e.zry abelian group is the direct product of its Sylow subgroups when- ever its order is divisible by more than one prime number. This implies that a necessary and sufficient condition that two abel- ian groups are simply isomorphic is that all their Sylow sub- groups are simply isomorphic, and hence the study of abelian groups is reduced to the study of such groups whose orders are powers of a single prime number. In particular, if the order of an abelian group is not divisible by the square of a prime number the group must be cyclic. Suppose that G is an abeUan group of order />", p being any prime number. If G is cyclic, all of its elements are generated by a single one of them si, where Sy may be selected in ^'" />""* ways. Moreover, every element of order p", m>a>0, in G is the pth power of p distinct elements of order p"'^^, the p^th power of p^ distinct elements of order p"'^'^, the p^'"~''hh power of p^'" distinct elements of order />*". When G is not cyclic we may choose for si any one of its elements of highest order. Some power of every other element of G is contained in the group (si) generated by si. We represent by S2' one of those elements which have to be raised to the highest power p"' to obtain an * A number of the theorems on abelian or commutative groups were first developed by L. Eulcr and C. F. Gauss in connection with their studies in number theory. The earliest extensive exposition of the properties of these groups is due to G. Frobenius and L. Stickclbcrger, Crelle, vol. 86, p. 217. 87 88 ABELIAN GROUPS (Ch. IV clement of (^i). That is, the p"th power of every element of G is in (51) but the />"'"' power of S2' is not in (si). We proceed to prove that S2' may be so selected that the cyclic groups (51) and {s'2) have only the identity in common; that is, Since 52''^ is in (^i), and the order of 52' does not exceed that of ^1, there must be an element in (si) whose p^^th power is the inverse of 52'*""'. The product of S2' and this element is therefore of order p'\ Moreover, the /?'~Hh power of this product is not contained in (si) since one of its factors is in (si), while the other does not have this property. In what follows we shall denote this product, of order p*, by 52. If the order of G is the product of the orders of Si and S2 it is clear that G is the direct product of (51) and (52). We proceed to prove that G is always the direct product of cyclic groups.* The orders of these cycUc groups are called the invariants f of G. In particular, if si is of order />" and if G is the direct product of (51), (52), the invariants of G are />"', />". If G contains elements which are not included in the group (51, S2), generated by si and ^2, we may suppose that the p*th power of every other element is in (51, S2) while the />"*"* th powers of some of the elements are not in this subgroup. We proceed to prove that at least one of the latter elements is of order P"*. Let S3' be any one of these elements. As is in (51, S2), and as a3<a2<ai, there must be some ele- ment in (^1, S2) whose />"'th power is the inverse of s^'""'. The product, 53, of this element and j'3 is therefore of order p"', and G involves the direct product of the three groups (^i), (^2), (sz)- As this process may clearly be continued until all * It is implied that each of these cyclic groups has only the identity in common with the group generated by all the others. In other words, they are independent cyclic groups. t The invariants of an abclian group have also been called the elementary divisors of the order of this group. Frobenius and Stickelberger, Crelle, vol. 86 (1878), p. 238. 35) INVARIANTS OF ABELIAN GROUPS 89 the elements of G have been exhausted, it has been proved that every noti-cyclic abelian group of order p^ is tlie direct product of independent cyclic groups. This is the most important theorem relating to abelian groups.* These independent cyclic groups may be represented as substitution groups on distinct sets of letters. Moreover, it is clear that a group can be constructed such that the orders of such substitution groups are arbitrary, and hence the prod- uct is an abelian group of arbitrary order. That is, the number of distinct abelian groups of order />" is equal to the number of the possible partitions of m as regards addition, and each of these groups may be completely defined by the value of p and the s>Tnbol (mi, W2, , Wx) where mi, m2, . . . , m\ repre- sent positive integers such that W1+W2+ . . . -\-m\ = m. The group G is completely defined by p and the values of the inte- gers mi, mo, , ffi\, and it is not affected by the order in which these integers are arranged. It is said to be of order p^ and of type (mi, W2, . , m-y). We may therefore suppose that the numbers mi, m2, - . . , mx are always arranged in order of magnitude, beginning with the largest. It may be added that the determination of the number of possible abelian groups of order p^ is reduced by these theorems to a problem in the theory of numbers; viz., the determination of the total num- ber of possible partitions of m as regards addition. This prob- lem has received considerable attention, but it still involves many unsolved difficulties. Suppose that an abelian group G of order /> has or invariants. All of its elements whose orders divide p must constitute a sub- group of order p"; and, conversely, whenever these elements constitute a group of order p", G has exactly a invariants. The * A set of independent generating elements can generally be selected in a large number of ways. Such a set is often called a base of the abelian group, and the operators si, Si, 5j, . . . are called elements of the base. The funda- mental theorem that every abelian group is a dircA product of independent cyclic groups is implicitly contained in the works of C. F. Gauss and K. Schering, but L. Kronecker gave the first satisfactory proof of it in 1870. It should periiaps be placed next after Cayley's theorem among the most fundamental theorems of group theory. 90 ABELIAN GROUPS [Ch. IV elements of G whose orders divide p^ constitute a group of order ^^**~^, where /3 is the number of invariants of G which are equal to p. In general, let Xi, X2, . . . , x^ represent the number of the invariants of G which are equal to p, p^, . . . , p^ respectively, and supp)Ose that they include all the invariants of G. The number N of the elements of G whose orders divide />*, /fe < X, is then given by the formula It should be observed that the orders of the independent cyclic groups which generate a given group G are completely determined by G when G is of order />"*. In general, G is the direct product of Sylow subgroups and hence it is also the direct product of a series of cyclic independent subgroups Ci, C2, . . . , Cm\ each of which has for its order a power of a prime number. Unless the contrary is stated it will be assumed that the order of each of these subgroups exceeds unity, and hence their number and their orders are completely determined by G; and, in turn, they determine G completely. That is, if these orders are the same for two groups these groups are simply isomorphic. These orders are therefore invariants of G, but they are not the only numbers which are known as invariants of G. Their number constitutes the largest possible number of orders of independent cyclic groups in G; that is, neither G nor any of its subgroups can have more than m' independent generators. It is important to note that the term set of independent gen- erators as regards abelian groups is usually employed to repre- sent a set of independent generators which is such that the group generated by an arbitrary number of them has only the identity in common with the group generated by the remaining ones. In dealing with abelian groups we shall always use this term with this special meaning and not with its general meaning given in 3. 36. Largest and Smallest Number of Possible Invariants. We proceed to find the smallest number of independent gen- erators of G. The given subgroups Ci, C2, , Cm' can be 36] NUMBER OF POSSIBLE INVARIANTS 91 arranged in rows such that the orders of all those in one row are powers of the same prime, and such that the order of each is equal to or greater than the order of the one which follows it in the same row. In case the rows do not contain the same number, the vacant places may be filled by the identity. Arranging these rows in the form of a rectangle, we have Cl, C2, . . . , Ca ^o+l> ^o+2> ^2a By forming the products of all those in each column we obtain a independent cyclic subgroups such that the order of each is divisible by the orders of all those which follow it. These subgroups form the smallest possible number of generating cyclic subgroups of G. The orders of these subgroups are com- monly called the invariants of G, since any other system of independent generating subgroups in which the order of every group is divisible by the order of every following group is composed of groups which are simply isomorphic with these products. It may be observed that a is the largest number of invariants in a Sylow subgroup of G, while m' is equal to the sum of the numbers of invariants of all the Sylow subgroups of G. It is clear that the independent generators of G can be so selected that their number has any arbitrary value from a to m' , but this number can have no other value. Moreover, G cannot be generated by less than a cyclic subgroups even if these subgroups are not independent. Whenever the independent generators of G are so chosen that the order of each of them is divisible by the orders of all those which follow it, their number must be a, and when the order of each is a power of a single prime their number must be m\ but it is not true that the independent generators can be so arranged that the order of each is divisible by the order of all those which follow it whenever the number of these generators is a. The two numbers a and m' are only equal when the order of G is a power of a single prime. Since the former method leads to the smallest number of invariants it seems appro- I 92 ABELIAN GROUPS [Ch. IV priate to call the orders of these independent generators the invariants of G, although the latter method has some advan- tages. The choice of invariants such that their number lies between a and m' seems less natural. Wc evidently arrive at the a invariants if we choose the independent generators in the following way. Start with an element of highest order and then select any other element such that the two generate the largest possible subgroup. The orders of two independent generators of this subgroup are the first two invariants of G. If we add to this subgroup another element so that the three generate the largest possible subgroup, we arrive at the third invariant, etc. A marked difference between the two given methods of arriving at the invariants of G should perhaps be emphasized; viz., the orders of the independent generators of G are completely determined by m\ but not by a. That is, if two sets of m' independent generating cyclic subgroups of G were given, the orders of the subgroups of one set would be the same as those of the other; but if two sets of a independent generating cyclic subgroups of G were given, the orders of those of one set could generally vary a great deal from the orders of those of the other. A necessary and sufficient condition that the orders of these two sets must be the same is that the a invariants of each of the Sylow subgroups of G, with one possible exception, are equal. The number a is said to be the rank of G. If G is the direct product of a series of subgroups Gi, G2, . . . , G\, we may select a set of independent generators of G by combining arbitrary sets of independent generators of each of these subgroups. Suppose that Gi, G2, . . . , G\ are the Sylow subgroups of G. Any element t oi G will have a constit- uent, which may be the identity, from each one of these sub- groups, and the order of / will be the product of the orders of these constituents. To determine the number of the elements of a given order in G it is only necessary to determine the num- ber of elements of a given order in each of the Sylow subgroups. That is, if the order of / is />i'"/>2'^ px"^ (p\, p2. . . . , px being prime numbers), the number of elements of G whose 37l NUMBER OF ELEMENTS OF A GIVEN ORDER 93 order is equal to the order of t is the product of the numbers of the elements of orders />i*, p2*, , px'^^ in the respective Sylow subgroups of G. We proceed to determine this number. 37. Number of Elements of a Given Order. Let G be any abelian group of order />*" whose invariants are />"', p*^, . . . , ^"^(ai f a2 > as > . . . >ax>0). Let- m'i = \ represent the number of invariants ^ p, m'2 the number of those ^ />2, . . . , and m'a^ the number of those =/>"*. To determine the num- ber of the elements of order />^(l^/3^ai), it is only necessary to find the order of the group generated by all the elements whose orders divide />" and to subtract from this number the order of the group formed by all the elements whose orders divide p^~^. That is, the number of elements of order p^ in G is equal to * To obtain the number of the elements of a given order n in any abelian group we may wTite n in the form 2'pi*p2'^ . . . Px"'^, and find the number of the elements of order 2** in the Sylow subgroup of even order, then find the number of the elements of order />i"' in the Sylow subgroup whose order is divisible by pi, etc. The product of all the numbers obtained in this way is equal to n. For instance, to find the number of the elements of order 12 in the abelian group whose invariants are 24, 6, 2, we observe that the invariants of the Sylow sub- groups are 8, 2, 2 and 3, 3 respectively. The number of the elements of order 4 in the former Sylow subgroup is 8, (m'l = 3, m'2 = 1, ^'3 = 1), and the number of those of order 3 in the latter is also 8. Hence there are exactly 64 elements of order 12 in the given abelian group. The number of the elements of order p^ in the group G of order />"* may also be obtained by observing that if 5i, 52, . . . , 5x represent a set of base elements of G, a set of base elements of the subgroup of G generated by all its elements whose HefTter, Crellc, vol. 119 (1898), p. 261; Netto, VorUsungen ilbcr Algebra, vol. 2 (1900), p. 248. h 94 ABELIAN GROUPS [Ch 1 orders divide ffi may be obtained by raising all those of the given set whose orders exceed p^ to a power sufficient to reduce their orders to p^. The elements whose orders divide p^'^ constitute a subgroup of index p'^'fi under the given subgroup of G. Hence the number of elements of order ^ is p^'fi\ times the order of the subgroup of G which is composed of all its elements whose orders divide /?^~^ EXERCISES 1. Determine the number of elements of each order in the four abelian groups of order 100. 2. The smallest number of letters on which an abelian group can be represented as a substitution group is the sum of its invariants, if all these invariants are powers of prime numbers. 3. A group of order p''q must be abelian when ^ is a prime number which is less than the prime number p and does not divide p-l. 4. If all the elements besides the identity of a group are of order 2 the group is abelian. 5. The order of the <^-subgroup of any abelian group of order g is equal to g divided by the order of the subgroup generated by all the operators of prime orders contained in this abelian group. 38. Abelian Groups of Given Orders. The number of the different possible abelian groups of order n = 2'^pi'^p2^ . . . p^"^ (pi, p2, , p\ being distinct odd primes) is equal to the product obtained by multiplying together the numbers which separately represent the total numbers of partitions, as to addi tion, of the separate exponents ao, ai, "2, ,ax which exceed zero. In particular, if none of these exponents exceeds 3 the number of distinct abelian groups of order n is equal to the product of those exponents which exceed zero. For instance, the number of abelian groups of order 32- 5^- 7-^ is 18, while the number of those of order 2'^ -32 -5^ is 7 2-5 = 70. From the preceding paragraph it results that it is very easy to determine the number of the possible abelian groups of a given order. It may be observed that two abelian groups which involve the same number of elements of each order are simply isomorphic, but this is not true as regards non-abeUan groups. In fact, we can easily construct non-abelian groups which have the same number of elements of each order as i 39] SPECIAL CI^SS OF ABELIAN GROUPS 96 certain abelian groups. For instance, if 5 is an element of order 8 while / is of order 2, the two groups of order 16 generated by s and t, when these elements satisfy one of the following equations tst=s, tst=s^, evidently contain the same number of elements of each order. To prove that two abelian groups which have the same number of elements of each order are always simply isomorphic, it is clearly only necessary to consider the case when their order is a power of a prime. In this case, it is easy to see that any change in the invariants will affect the number of elements of given orders. In fact, if the order of such a group is p"*, it has been observed that the elements of order p generate a group of order p, where a is the number of its invariants. The elements of order p^, if there are such, generate a group of order p^~^ where ^ is equal to the number of invariants which are equal to p, etc. Hence it results that two abelian groups which have the same number of elements of each order are simply isomorphic. 39. A Special Class of Abelian Groups. We gave, in 5, illustrations of abelian groups which are generated by the 0(g) totitives of g, that is, by the <f>{g) natural numbers which do not exceed g and are prime to g, and we shall now enter upon a more detailed study of these important abelian groups.* It is easv to s ee that they c onstitute the groups of isomorphisms of cyc lic groups ; that is, the groups according to which the elements of cyclic groups are permuted when these cyclic groups are made simply isomorphic with themselves in every possible manner. For the sake of simplicity we begin with two illus- trative examples. Let 1, a, o?, o-"^, a^, represent the five fifth roots of unity. These may be put into a (1, 1) correspondence, or they can be made simply isomorphic with each other, in the following four ways, but in no other way: * In the second edition of vol. 2 of Weber's Lekrbuch der Algebra, 1809, j.. (,(), these groups are called the most important example of abelian groups of finite order. 96 ABELIAN GROUPS (Ch. IV 1 1 1 1 1 1 1 1 aa aa2 ac? a a"* o^o^ a2a4 o?a o?o? o?o? o?a o^a^ c?o? o^c^ a'^o?. o^o? o^a As the different ways in which a group can be made simply isomorphic with itself clearly correspond to the substitutions of a group, it results that the group of isomorphisms of the group of order 5 is of order 4. As a substitution group whose elements are the numbers 1, 2, 3, 4 the given group of isomorph- isms may be represented in the following manner: 1, 1243, 1342, 14. 23. That is, the group of isomorphisms of the group of order 5 is the cyclic group of order 4. The same abstract group of order 4 is evidently generated by the following four numbers, when they are combined by multiplication: ^ 12 3 4. (mod 5); As a second illustrative example we consider all the possible simple isomorphisms between the eight eighth roots of unity represented by the following symbols: 1, /3, /S^, /3^, ^, j8^, /3^, /3", where ^ is any root of the equation ic* + 1 = 0. isms may be represented as follows: 1 1 1 1 |8/3 /3/33 /32/32 /32/36 /33^ /33/3 ^^ ^/3* fi^0^ /3S^7 jge/S* ^6^2 ^7^7 ^7^5 ::4 + l=0. These isomorph- 1 1 1 1 ^/3^ /3^7 ^2^2 /32^6 /3307 /33/35 ^^ ^^ /3/3 ^5^ ^6^6 ^6^2 /37^ ^70 This group of isomorphisms, as a substitution group on the numbers 1, 2, . . . , 7, is as follows: 1 13-57-26 15-37 17-35-26. y 39] SPECIAL CLASS OF ABELIAN GROUPS 97 This group could have been equally well represented by 1 13-57 15-37 17-35 and it is clear that it is simply isomorphic with the group formed by the following numbers, when they are combined by multi- plication : 13 5 7 (mods). These examples may sufl&ce to illustrate the fact that the group formed by the <^(w) totitives, with respect to multiplica- tion (mod w), is the group of isomorphisms of the cychc group of order m. To prove this fact it is only necessary to observe that the correspondence of the operators of lower orders in a cyclic group is completely determined by the correspondence of the operators of highest order, and that all of the latter may be obtained from any one of them by raising it to all the various powers which are prime to the order of the cyclic group. In particular, a necessary and sufficient condition that the num- ber m has a primitive root is that the group formed by the <^(w) totitives (mod w) be cyclic. While the group formed by the totitives is always abelian, there are many abelian groups which cannot be represented in this way. Hence these groups form a special class of abeUan groups. We proceed to determine some conditions which must be satisfied in order that an abelian group G may belong to this class. When G is cyclic the matter is quite simple. It is necessary and sufficient that its order g be the exponent to which a primitive root of some number belongs. That is, whenever g = p^{pl),* ^ being an odd prime number, and a being any positive integer or zero, the cyclic group G belongs ) the given class, and only then. The lowest two even num- >v.'rs which are not of the form />"(/> 1) are 8 and 14; hence these numbers are the lowest orders of cyclic groups of an even order that cannot be the groups of isomorphisms of any cyclic groups whatever, and hence the cyclic groups of these two orders cannot be represented as groups of totitives. * Cf. Dirichlet-Dedekind, Zahkntheorie, 1879, p. 340. : ABELIAN GROUPS [Ch. IV If g is written in the form g = 2"/>i'"/>2' . {pi, p2, . - being different odd prime numbers); G is the direct product of its subgroups of orders 2**, pi', p2*, . . , and its group of isomorphisms / is evidently the direct product of the groups of isomorphisms of these subgroups. Since the group of iso- morphisms of a cyclic group whose order is a power of an odd prime number is cyclic, it follows from the above that / is the direct product of the cyclic groups of orders />]"" *(/>i l), />2***~*(/>2 1), , when 00 = or 1. When ao>l, we have to add a group of order 2 and a cyclic group of order 2'^'^ to these factor groups in order to obtain /, since there are numbers which belong to the exponent 2*^"^ (mod 2"), but none which belong to a higher exponent.* Since / is the direct product of groups of even orders, the order of / is always even when g>2. It can clearly be any even number of the form 2^pi^'p2^ . . . {pi-l){p2-l) . . . The smallest two natural numbers which are not of this form are 14 and 26 ;t hence these numbers are the lowest orders of groups that cannot be groups of isomorphisms of any cyclic group whatever. It is evident that the highest prime factor of the order of / can not exceed the highest prime factor of g. EXERCISES 1. Determine the invariants of the group formed by the 40= (^(100) totitives of 100 (mod 100). 2. The number of invariants in the group of the totitives of m (mod m) is equal to the number of the distinct odd prime factors of m when- ever m is either odd or double an odd number. It is equal to the num- ber of distinct prime factors of m, whenever m is divisible by 4 but not by 8; when m is divisible by 8, the number of these invariants is one more than the number of the distinct prime factors of m. Cf. Weber, Lehrbuch der Algebra, vol. 2, 1896, p. 59. 3. Find the three possible cyclic groups whose group of isomorphisms has the invariants 6, 2, 2. 4. If the operators of order 2 in the group of isomorphisms of the cyclic group of order m generate a group of order 2", what is the maximum num- ber of the distinct primes which divide w? What is the minimum number of such divisors of m? * Cf. H. VVcbcr, Lehrbuch dcr Algebra, 2d cd., vol. 2, 1899, o. 64. t Lucas, Thiorie dcs nombrcs, 1891, p. 394. 40] SUBGROUPS AND QUOTIENT GROUPS 99 5. If the group of totitives of m has for its order a power of a prime this order is of the form 2". 6. Every p)ossible abelian group is a subgroup of some group of totitives. Suggestion: If the invariants of the given abelian group are so chosen that each is a power of a prime number, it is clearly possible to choose m so that the group of totitives of w involves the same invariants. 40. Subgroups and Quotient Groups of any Abelian Group. It has been proved that every abelian group may be regarded as the direct product of cyclic groups and hence it is completely determined by the orders of these groups. As every subgroup of an abehan group is abeUan, it results that these subgroups are also completely determined by the orders of the cycUc groups of which they are the direct products. Hence it follows immediately that a necessary and sufficient condition that an abelian group G whose invariants are i\, i2, . . . , i^ contains a subgroup whose invariants are ji, J2, , jt is that it be possible to associate the tj's with t distinct i's so that each i is equal to or greater than the corresponding j . If such an arrangement were not possible the subgroup would involve more operators of a certain order than the entire group. The condition imposed upon the invariants of a sub- group is clearly equally applicable as regards the invariants of a quotient group. Hence we have the important theorem: The invariants of any subgroup of an abelian group are invari- ants of a quotient group, ar^d the invariants of any quotient group are also invariants of a subgroup. In other words, each subgroup is simply isomorphic with a quotient group and vice versa. A like theorem is not always true as regards non-abelian groups. If a group is cyclic all of its subgroups may be obtained by raising successively all of its operators to the same power, but this method cannot give all the subgroups of a non-cyclic group. The ^th power of each operator of an abelian group G gives a group which is simply isomorphic with (7 whenever k is prime to g. If k is not prime to g, these ^th powers con- stitute a quotient group of G, whose order is g divided by the total number of the operators of G whose orders divide k. 100 ABELIAN GROUPS (Ch. IV While it is not diiEcult to find, by means of the theorem stated above, the total number of the different types of sub- groups in a given abelian group whose invariants are known, it is a problem of considerable difficulty to determine all the pos- sible subgroups of the same type. To illustrate this fact we consider the subgroups of the important class of abelian groups of order p^ and of type (1,1, . . . , to m units). In this case there are evidently m \ different types of subgroups, exclud- ing the identity. That is, there is one and only one type of subgroups of order />, a = l, 2, . . . , w 1, separately. In this case it is also not very difficult to determine the number of the different subgroups of order />". In fact, this number is clearly equal to the quotient obtained by dividing the total number of ways in which generating operators of such a subgroup can be selected from the operators of the group by the number of ways in which such generators can be selected from the operators of the subgroup. Hence this number is {r-^)ir-p)if^-f) ir-p"-') (P^-iKP^-PHP^-P) . ip^-p"-') (r-i)(r-^-i) (r -a+l 1) (/^-l)(/>--l) (P-I) In the particular case when a = fn l, this formula reduces to p-1' Hence there are as many subgroups of order />""' in an abelian group of order p"* and of type (1,1, 1, . . . ), as there are sub- groups of order p. For instance, the group of order 8 and of ty-pe (1, 1, 1) has seven subgroups of order 2 and also seven subgroups of order 4, while the group of order 16 and of type (1, 1, 1, 1) has fifteen subgroups of order 2 and fifteen subgroups of order 8, 4ll ISOMORPHISMS OF AN ABELIAN GROUP 101 EXERCISES 1. A necessary and sufficient condition that a group be abelian is that each operator corresponds to its inverse in one of the possible auto- morphisms of the group.* 2. Find the number of subgroups with invariants 6, 2 in the abelian group whose invariants are 12, 6, 2. 3. Determine the number of the subgroups of each p)ossibIe order in all the abelian groups of order />', p being a prime. 4. Every abelian group is generated by its operators of highest order. 5. Give an instance of a non-abehan group which is not generated by its operators of highest order. . 41. Group of Isomorphisms of an Abelian Group, f Some of the most useful properties of an abelian group are exhibited by its group of isomorphisms. We have already considered the group of isomorphisms of a cyclic group and found that it is an abelian group. We shall see that a necessary and sufficient condition that the group of isomorphisms of an abelian group G be abelian is that G be cyclic, and hence it results that the groups of totitives are the only abelian groups of isomorphisms of abelian groups. This fact is a special case of the theorem that the invariant operators of the group of isomorphisms of any abelian group constitute a group which is simply isomorphic with the group of the totitives of^the largest possible invariant of this abelian group. For instance, if an abelian group has the invariants 10, 10, 2, the invariant operators of its group of isomorphisms constitute the cyclic group of order 4. We pro- ceed to prove the stated theorem. We shall first prove that if an operator / of the group of isomorphisms of an abelian group G transforms every operator of G into the same power (7th) of itself it must be commutative with every operator of the group of isomorphisms of G. Let ti be any other operator of this group of isomorphisms and sup- pose that This includes the theorem that every group which invdves no operator whose order exceeds 2 is abelian. t Cf. A. Ranum, Transactions of the American Mathematical Society, voL 8 (1907), p. 83. 102 ABELIAN GROUPS [Ch. IV Sa being an arbitrary operator of G. Since it results that / and h are commutative. On the other hand, suppose that / is commutative with every operator in the group of isomorphisms of G. We shall first prove that / must transform every operator of highest order in G into a power of itself. For, if Sa is such an operator and where 5a is not a power of Sa, it is clearly possible to find an operator /i in the group of isomorphisms of G such that h is commutative with 5 but not with s^. As it is necessary then that it results that / is not commutative with every operator of the group of isomorphisms of G unless it transforms every operator of highest order of G into a power of itself. We shall now show that t must transform into the same power every operator of highest order in G, and hence it must transform every operator of G into this power, since these opera- tors of highest order generate G. It results from the manner in which the invariants of any abelian group were determined that the group of isomorphisms of G transforms its operators of highest order transitively. That' is, the group of isomorph- isms of G may be represented as a transitive substitution group in which each letter stands for an operator of highest order in G. If Sa, s^ represent two operators of highest order in G and if we can find an operator tz in the group of isomorphisms of G such that Hence it results that That is, /, /2 are not commutative unless 7 = 5. This com- pletes a proof of the theorem: A necessary and sufficient con- 41] ISOMORPHISMS OF AN ABELIAN GROUP 103 dition that an operator of the group of isomorphisms of an abelian group be invariant under this group, is that it should transform every operator of this abelian group into the same power of itself* Since two abelian groups having the same invariants can be made simply isomorphic, and two simply isomorphic groups have the same invariants, it results that the order of the group of isomorphisms expresses also the number of different ways of choosing the independent generators of the group. It should be observed that while every operator of highest order in an abelian group may be used as an independent generator, and hence each operator of highest order must correspond to every other operator of this order in some simple isomorphism of the group with itself, it is not generally true that every operator of lower order corresponds to every operator of its own order in some simple isomorphism of the group. This fact may be illustrated by means of the abelian group whose invariants are p^ and p. It is evident that this group contains a characteristic subgroup of order p; viz., the subgroup of order p which is generated by its operators of order p^. The remaining p subgroups of order p in the given group of order p^ are conjugate under the group of isomorphisms of this group. In any automorphism of any abeUan group G each operator of G corresponds to itself multiplied by some operator of G. The totality of these multiplying operators evidently consti- tutes a group T which is either G itself or a subgroup of G, and the automorphism may be obtained by making G isomorphic with T and multiplying corresponding operators. In this isomorphism no operator except the identity can correspond to its inverse. As this condition is necessary as well as sufficient we have arrived at the following fundamental Theorem: Every automorphism of an abelian group G may be obtained by (1) making G isomorphic with one of its subgroups or with itself in such a manner that no operator besides the identity corresponds to its inverse, and (2) making each operator of G * Transactions of the American Maihematical Society, vol. 1 (1900), p. 397; vol. 2 (1901), p. 2G0. 104 ABELIAN GROUPS [Ch. IV correspond to itself multiplied by the operator which corresponds to it in this isomorphism. The simplest case that can present itself is the one in which the subgroup of G which corresponds to the identity of T includes T. The resulting simple isomorphism of G with itself must correspond to an operator in the group of isomorphisms of G, whose order is equal to the order of the operator of highest order in T. When the order of T is an odd prime number p, or double such a number, only one other case can present itself; viz., the case in which T, or its subgroup of odd order, corresponds to itself in the given isomorphism between G and T. In this case the isomorphism corresponds to an operator whose order divides p \, in the group of isomorphisms of G. These results give rise to the following theorem: // we make an abelian group G simply isomorphic with itself by multiplying its opera- tors by those of a subgroup whose order is p, or 2p (p being an odd prime), the resulting automorphism of G corresponds to an opera- tor of order p, 2p, or {p l)/a {a being a divisor of p I), in the group of isomorphisms of G. The determination of all the possible orders of the corre- sponding operators in the group of isomorphisms of any abelian group, when T is a given subgroup, seems to be a problem of considerable difficulty. When the order of T is small the num- ber of cases that have to be considered is also small. In addi- tion to the orders included in the given theorems, we have the following, when the order of T does not exceed 8: If T is the cyclic group of order four, the resulting isomorphism may also correspond to an operator of order two in the group of isomor- phisms, and when T is the non-cyclic group of this order, it may also correspond to operators of orders 3 and 4. When T is the cyclic group of order 8, the orders of these operators may be 2, 4, and 8; when T is the direct product of the cyclic group of order 4 and an operator of order 2, the orders of the corresponding operators in the group of isomorphisms may be 2 and 4; finally, when T is the direct product of three operators of order 2, the given operators may be of orders 2, 3, 4, 6, and 7. While all of the possible cases for a given T may present themselves in 42] ISOMORPHISMS OF THE GROUPS OF ORDER p^ 105 the same group, it is evident that this does not always happen. For the sake of illustration we consider the group of iso- morphism of the group of order 8 which is the direct product of three operators of order 2* Each of its 7 subgroups of order 4 leads to three operators of order 2. We thus obtain the 21 operators of order 2 of the required group of isomor- phisms when we consider all the possible instances in which the order of T is 2. If the order of T is 4, two cases present themselves, in one case just two of the operators of T (includ- ing identity) correspond to operators of T, and in the other case each one of the operators of T corresponds to some opera- tor of T. The former case leads to the 42 operators of order 4, and the latter to the 56 operators of order 3 of the required group of isomorphisms. Finally, we obtain 48 operators of order 7 when we consider all the possible instances in which the order of T is 8. Hence the group of isomorphisms in question is the well-known group of degree 7 and of order 168. 42. Groups of Isomorphisms of the Groups of Order p^. A group of order p^ is abelian and there are two such groups. The group of isomorphisms / of the cyclic group of order p^ is the cyclic group of order p{p \). If 5 is a generator of this group of order p^, we may select for a generator of its / any operator / of order p{p l) such that t-^st = s", where a is any primitive root of p^. If the order of the multiplying subgroup T in such an isomorphism is p, the corresponding operators of / constitute its subgroup of order p. The remaining operators of I result when T is of order p^, p being an odd prime number. When the group G of order p^ is non-cycUc, the order of its / is clearly {p^l){p^ p), and this / is isomorphic with a tran- sitive group of degree p-\-l corresponding to the transformations of the />-(- 1 subgroups of order p contained in G. There is clearly a (^ 1, 1) isomorphism between / and this transitive group of degree p-\-l, since the p l invariant operators of / are the only ones which transform every subgroup of G into itself. * C{. E. H. Moore, Bulletin of the A merican Mathematical Society, vol. 1 (1894), p. 63. 106 ABELIAN GROUPS [Ch. I\ If we regard / as a substitution group on p^\ letters it is transitive, its subgroup composed of all its substitutions which omit one letter omits p \ letters, and it is a regular group on the remaining p^ p letters. This subgroup contains a single Sylow subgroup of order p, which corresponds to the cases when the multiplying subgroup T is composed of the invariant operators of G under the isomorphisms in question. The orders of all its other operators divide /> 1 as they corre- spond to the cases when T, under the isomorphism in question, is a non-invariant subgroup of order p; and this subgroup of order p{p l) is simply isomorphic with the metacyclic group of degree p, which represents the transformations of p Sylow subgroups of G under this subgroup of /. As a substitution group on p^l letters the group / is clearly imprimitive when p is odd, and its />-}-l systems of imprimitivity are permuted according to a doubly transitive group of degree />+l and of order p{p^ l), discovered by Mathieu.* It is easy to determine the number and the orders of all the substitutions of / whose degrees are less than p^ 1. In fact, since the subgroup of 7 which is composed of all its sub- stitutions omitting one letter is regular, and of order p'^p, it follows directly that I involves {p'^p\){p-\-\) different substitutions each of which omits exactly p\ letters. The order oi pf^ \ of these is p, while the orders of all the others are divisors of /> 1. li d represents any divisor of p \, then the number of these substitutions which are of order d is p{p-\-\)<i>{d). All the substitutions of order p are conjugate, but there are <t>{d) equal sets of conjugate substitutions of order d and of degree p'^p. The isomorphic group of degree />+ 1 and of order p{p^ l), according to which the p-^1 sub- groups of G are transformed, contains also p^1 conjugate substitutions of order p, and all its substitutions whose degree is less than p are of degree pl and have orders which divide pl. The number of these substitutions whose order is d is hP(p-\-l)<t>{d). E. Mathieu, Paris Comptes Rendus, vol. 47 (IfViS), p. 698. 43] CONFORMAL GROUPS " 107 43. Abelian Groups which are Conformal with Non-abelian Groups. Two distinct groups are said to be conformal when they contain the same number of operators of each order.* We proceed to determine all the abelian groups which are con- formal with non-abelian groups. The complete solution of the converse of this problem, viz., the determination of all the non-abelian groups which are conformal with abelian ones, is much more difficult, since a large number of distinct non- abelian groups may be conformal with the same abelian group, while no more than one abelian group can be conformal with one non-abelian group. In fact, it was observed in 37 that two distinct abelian groups cannot be conformal. It is evident that there is only one group of order 2"* which does not include any operator of order 4, viz., the abelian group of type (1, 1, 1, ,..). Moreover, there is only one cyclic group of order 2", and when w<4 no two groups of order 2^ are conformal. We proceed to prove that every abelian group G of order 2"* which does not satisfy one of these conditions is conformal with at least one non-abelian group. Let H be the subgroup of G which is generated by the square of one of its independent generators s of lowest order, together with all the other independent generators of G. The order of H is 2"*"^ Since w>3 there is an operator / of order 2 which has the following properties: It transforms H into itself, it is commutative with half of the operators of 11 (including all those which are not of highest order), and it transforms the rest into themselves multiplied by an operator of order 2 which is not the square of a non-invariant operator of //; i.e., / does not transform an operator of order 4 contained in // into its inverse. The non-abelian group generated by H and / is con- formal with G whenever s- = \. When the order of s exceeds two, the group generated by / and H (written as a regular substitution group) may be made simply isomorphic with itself by writing it on two dis- tinct sets of letters. If in this intransitive group / is replaced by the continued product of /, the substitution of order two Quarterly Journal of Mathematics, vol. 28 (1896), p. 270. 108 ABELIAN GROUPS [Ch. IV which merely permutes corresponding letters of the two sys- tems of intransitivity, and s^ in one of the systems of letters, there results a transitive group which is conformal with G. That is, any abelian group of order 2, m>S, which is neither cyclic nor of type (1, 1, 1, ...), is conformal with at least one non-abelian group. It will now be assumed that the order of G is p"* {p being an odd prime number and w>3), and that G is non-cyclic. Let H be the subgroup generated by s^ {s being one of the indepen- dent generators of lowest order in G) together with all the other independent generators of G. There is an operator / of order P which transforms H into itself, is commutative with each of its operators contained in a subgroup of order />'""^, and transforms the rest into themselves multiplied by invariant operators of order p. This t and // generate a group which is conformal with G whenever 5" = I; for, if si is any substitution of H that is not commutative with /, it is easy to see that {tsiY = tSitS\ . . . (/> times) = tsit-Hhit-Hhit-'^i* . . . /^-''/''5i=5i''.* When 5" differs from identity the group generated by H and /, written as a regular group, may be made simply isomorphic with itself p 1 times, by writing each substitution in p dis- tinct sets of letters; and / may be replaced by the continued product of /, the substitution of order p which merely permutes the corresponding letters of these systems of intransitivity, and the pth power of s in one of these systems. In the result- ing group the pth power of the operators will be the same as those of G taken in the same order, and hence this group will be con- formal with G. If a non-abelian group whose order is not some power of a prime is conformal with an abelian group G, it must be the direct product of its Sylow subgroups, and hence each of these sub- groups is conformal with an abelian group, and at least onei of them is non-abelian. From what precedes, it may be observed that necessary and sufficient conditions that any abelian group * Transactions of the American Mathematical Society, vol. 2 (1901), p. 262. 44] CHARACTERISTIC SUBGROUPS 109 of order Q^'pi'^po"* . . . {pi, p2, . . . being distinct odd primes) be conformal with at least one non-abelian group are: 1^ at least one of its subgroups of orders 2, px\ P'f^ ... is non-cyclic; 2^ if the order p^& of this subgroup is odd, then a^> 2; if the order is eVen (2), then the subgroup must involve operators of order 4 and ao>3. Since any number of these factors may be non-abelian, there cannot be an upper limit to the number of non-abelian groups which can be conformal with an arbitrary abelian group. This fact may be seen in many other ways. EXERCISES 1. Let s be of order 16 and let / represent an operator of order 2 such that tst=s^\ prove that (5, /) is conformal with the abelian group whose invariants are 16, 2. 2. Find the group of isomorphisms of the abelian group whose invari- ants are 8, 2 and determine its invariant operators. 3. If p is an odd number, there is at least one non-abelian group of order p^^ m>2, which is conformal with the abelian group of type (1, 1, ... to 7M units). The number of such possible groups increases with m and has no upper limit. 4. Any operator of order />* in any abelian group whatever can be used as an independent generator provided its />*~^th power is not included in a cyclic subgroup of order />*"^^. 5. Every jxjssible group of finite order is a subgroup of the group of isomorphisms of an abelian group of order 2"* and of type (1,1,1,...). Suggestions: Observe that this group of isomorphisms contains a subgroup which is simply isomorphic with the symmetric group of degree m. 44. Characteristic Subgroups of an Abelian Group. In 29 a characteristic subgroup was defined as a subgroup which cor- responds to itself in every possible automorphism of the group. An operator which corresponds to itself in every possible automorphism of the group is Ukewise called a characteristic operator. It is clear that every characteristic subgroup is also an invariant subgroup, and that every characteristic operator is also an invariant operator; but invariant subgroups and invariant operators are not necessarily characteristic. The Sylow subgroups of an abelian group whose order is not a power of a single prime are evidently characteristic subgroups. 110 ABELIAN GROUPS [Cn. IV If G is an abelian group of order ff* and of type (1,1,1, . . . ), it contains no characteristic subgroup besides the identity; but every other abelian group contains at least one character- istic subgroup besides the identity. Suppose that G is an abelian group of order />" but not of type (1, 1, 1, ... ), and let />* be one of its largest invariants. If exactly Xi of the invariants of G are equal to />"', then G contains a characteristic subgroup of order p^' and of type (1, 1, 1, ...). This characteristic subgroup Ci has been called the fundamental characteristic subgroup * of G, since it is contained in every possible char- acteristic subgroup of G besides the identity, as we shall prove in the following paragraph. It is evident that the subgroup of order p^\ which is com- posed of the identity and of all the operators of order p which are generated by the operators of highest order contained in G, is the characteristic subgroup Ci. Moreover, the conjugates under the group of isomorphisms / of every operator of order p, which is found in G but not in Ci, generate a characteristic subgroup of G which includes Ci as a subgroup. This fact results immediately from the different possible ways of select- ing the independent generators of G. If the second largest invariants of G are p"*, and if there are exactly X2 such invari- ants in G, then G contains also a characteristic subgroup C2 of order />^'+^, which is composed of the identity and of the operators of order p , which are generated by the operators! of order p* contained in G. By continuing this process w clearly arrive at the following Theorem. // a group G of order />" has Xi invariants whicH] are equal to p', X2 which are equal to />"*, . . . , X^ which are\ equal to pfi, where a\>a2> . . . >ap, then G has ^ character-l istic subgroups Ci, C2, , C^, besides the identity, such thati each of them is generated by operators of order p. Their orders] are p^\ />^'+^, , p^'+^+ +H, respectively, and each is\ included in all of those which follow it. Since every operator of highest order contained in an abelianj group can be used as an independent generator of the group,] * American Journal of Mathematics, vol. 27 (1!K).')), p. 15. 441 CHARACTERISTIC SUBGROUPS HI it is clear that a characteristic subgroup of G cannot involve any of its operators of order />"'. All the operators of G whose orders divide p'~^', where /3i has any value from 1 to ai 1, constitute a characteristic subgroup of G. The character- istic subgroup Cff of the preceding theorem corresponds to the case when /3i=ai 1. If all the invariants of G are equal, there is only one characteristic subgroup of G, besides the identity, which involves operators of order p^^ but none of higher order. It is also evident that the p^'th. powers of all the operators of G constitute a characteristic subgroup of G. If 5 is any independent generator of G, the conjugates of 5 under the group of isomorphisms of G generate a group which involves all the operators of G whose orders do not exceed the order of s. In other words, if a characteristic subgroup in- volves an independent generator of an ahelian group, it also involves all the operators of this ahelian group whose orders divide the order of this independent generator. This theorem clearly includes the theorem stated above, to the effect that a char- acteristic subgroup cannot involve any of the operators of highest order contained in G. In the following paragraph we shall estabUsh a still more general theorem in case />> 2. For a study of the special properties of the characteristic subgroups it is convenient to let Hi, H2, . . . , H^ repre- sent the subgroups of G which are generated respectively by a set of Xi independent generators of order />", a set of X2 independent generators of order p^, . . . , a set of X^ indepen- dent generators of order p". Suppose that p>2 and that s\ is some operator of order />*, a\> 5, which is contained in G, If ay> b^Uyj^i, and if S\ is the product of an operator of highest order in Hy+i and an operator of order p^ from Hy, then the conjugates of S\ under / generate a group which involves all the operators of order />* that are contained in G. In fact, this group clearly involves an independent generator of Hy^x since p>2, and it involves all the operators of order p* in the direct product of the subgroups Hi, ... , Hy. By means of the preceding theorems it is not difficult to determine the characteristic subgroups of any given abelian 112 ABELIAN GROUPS [Cm IV group of order p^. *If the order of an abelian group is not a power of a single prime number its characteristic subgroups are found by forming the direct product of the characteristic subgroups of its Sylow subgroups, and all such direct products are characteristic subgroups of G. EXERCISES 1. If an abelian group G of order />"" has only two distinct invariants pcn^ pat^ and if aiai = n, then the number of the characteristic subgroups which are generated by the operators of order p^ is 2, when =1 and h has any one of the values from 1 to ai 1, The number of these subgroups cannot exceed the smaller of the two numbers M-f 1, aj + 1 for any value of . 2. Find all the characteristic subgroups of the abelian group of order / and of type (1, 2, 3). 3. The abelian group of order 16 and of type (1, 3) has the propert\- that no single set of operators of order 4, which are conjugate under its group of isomorphisms, generates all its operators of this order. Prove that whenever p>2 all the operators of order p'^ in the abelian group of order p* and of type (1, 3) are generated by a single set of operators of order p- which are conjugate under its /. 45. Non-abelian Groups in which Every Subgroup is Abelian. Let G represent any non-abelian group all of whose subgroups are abelian. As instances of such groups we may cite the octic and quaternion groups. We shall first prove that G must contain an invariant subgroup of prime index p. Suppose that G is represented as a transitive substitution group of the smallest possible degree. If this group is imprimitive it must transform a set of systems of imprimitivity according to a prim- itive group which has a (1, a) isomorphism with G. This primitive group must be such that each of its subgroups is abelian, and hence we have only to prove that every prim- itive group which contains only abelian subgroups has an invariant subgroup of index p. If this primitive group were regular it would be of order p. If it were non-regular and of degree , a maximal subgroup of degree w 1 would be abelia and hence all of its substitutions besides the identity would of degree 1 ; for, if the degree of such a substitution were less than w 1 , this substitution would occur in two maximal i 45] GROUPS CONTAINING ONLY ABELIANSUBGROUPS 113 abelian subgroups and hence it would be invariant under the entire group. Consequently this primitive group of degree n would involve exactly 1 substitutions of degree n, while each of its remaining substitutions besides the identity would be of degree 1. The substitution groups which have these properties have been studied extensively. Frobenius * proved that such a group must have an invariant subgroup of order n. This im- portant fact will be proved in 139. If we assume this theorem for the present, it results that G must contain an invariant subgroup of index p, since the quotient group with respect to the given subgroup of order n is abelian. We shall now prove that the order g oi G cannot be divisible by more than two distinct prime numbers. Supp>ose that g = pj^p2* . . . px\, where />i, />2, . . . , /x are distinct prime numbers. Since G contains an invariant abelian subgroup of prime index, we may suppose that it contains an invariant subgroup H of order h, where h is given by the formula A = />i"'-V2"' . . . px"^- As H is the direct product of its Sylow subgroups and as every operator of G which is not in H has an order which is divisible by pi, it results directly that G contains only one Sylow sub- group of each of the orders p2% , p\">'- Let s be any operator, which is found in G but not in H and whose order is of the form //. As 5 transforms each of the Sylow subgroups of H into itself, and as G is non-abelian, J must be non-commutative with some of the operators in one of these Sylow subgroups. This Sylow subgroup and s must generate G, otherwise G would involve a non-abehan subgroup. This completes the proof of the fact that the order of a non- abelian group which contains only abelian subgroups cannot be divisible by more than two distinct prime numbers. Suppose that the order of G is />i"/>2"', a2>0, and that G contains an invariant subgroup of order />i'"*/>2"', and represent the Sylow subgroups of orders pi^ and p-z"* by Pi and P2 respec- * Frobenius, Berliner SUzungsberickte, 1902, p. 455. b 114 ABELIAN GROUPS [Ch. IV tively. We proceed to prove that Pi is cyclic and that P2 is of type (1,1, 1, ...). That Pi is cyclic follows directly from the fact that s transforms P2 into itself and that G is generated by s and P2. In fact, it is evident that ai=/3. If P2 were not of type (1, 1, 1, . . . )> it would contain characteristic subgroups generated by its operators whose orders are divi- sors of />2, p2^, . . , p2^~^, where P2'' is the order of its opera- tors of highest order. All of the operators of these character- istic subgroups would be composed of operators which would be commutative with s, since G cannot contain a non-abelian subgroup. Hence s would have to transform among them- selves all the operators of order P2'' in P2 which have the same p2th power. As the number of these operators is a power of p2, this is impossible. That is, we have arrived at an absurd- ity by assuming that r> 1, and hence we have established the theorem: If a non-abelian group which contains only ahelian subgroups has more than one Sylow subgroup, one of these sub- groups is of the type (1, 1, 1, ... ) and the others are cyclic. EXERCISES 1. If all the subgroups of a non-abelian group of order />*", p being a prime number, are abelian, its commutator subgroup is of order p and the pih power of each of its operators is invariant.* 2. Every subgroup of the dicyclic group of order Ap is abelian. 3. If every subgroup of order p"^~^ in a non-abelian group of order />"* is abelian, there must be exactly p-\-\ such subgroups. 4. The group of order 56 which contains 8 subgroups of order 7 does not involve any non-abelian subgroup. 46. Roots of the Operators of an Abelian Group. If 5i, 52, . . . , Sg are the operators of an abelian group G, and if G con- tains two operators 5, s^ which are such that Sa=Sfi, then s^ is said to be an th root of s^. In particular, every operator of G is a gth root of the identity, so that the identity has g gth roots under G. If n is prime to g every operator of G has one and only one nth root. On the other hand, if w is a divisor of g, the total number of the operators of G whose orders divide * Cf. G. A. Miller and H. C. Moreno, Transactions of the American Matlte- matical Society, vol. 4 (1903), p. 403. 47) HAMILTON GROUPS 116 n is always divisible by n. These operators constitute a sub- group III of G. If an operator of G has one th root it must have //I such roots, hi being the order of Hi. Whenever n is a divisor of g all the operators of G may be di\'ided into two classes according as they have nth roots or do not have this property. As each operator of the first class has exactly hi nth roots, the number of the operators in this class is g/hi. The number of the operators in the second class is there- fore g-g/hi. When G is cyclic, hi=n. If d is the highest common factor of n and g every operator of G which has one th root must have either d or a, multiple of d such roots, since the total number of the operators of G whose orders di\dde d is divisible by d. The number of the operators which have th roots is evidently equal to g/h2, A2 being the order of the subgroup H2 which is composed of all the operators of G whose orders divide d. The h2 operators of G which are th roots of the same operator correspond to the same operator in the quotient group G/H2. Every operator of G is clearly an wth root of one and of only one operator of G, but a given operator may have a number of different th roots. If a group is non-abelian two operators which have th roots need not have the same number of such roots. For instance, in the symmetric group of order 6 the identity has three square roots while each of the two operators of order 3 has only one such root. 47. Hamilton Groups. A non-abelian group G is said to be a Hamilton group, or a Hamiltonian group, if each of its subgroups is invariant. WTiile these groups are not abelian, they have in common with the abelian groups the property that every subgroup is invariant. They were thus named by Dedekind in honor of Sir W. R. Hamilton, and some of their fundamental properties were first studied by Dedekind in view of their usefulness in the study of the number reahns which belong to such groups.* Since every subgroup of G is invariant, G must be the direct * R. Dedekind, Mathematische Annalcn, vol, 48 (1897), p. 548. 116 ABELIAN GllOUPS (Ch. IV product of its Sylow subgroups. // all the subgroups of a Sylow subgroup of odd order are invariant, this Sylow subgroup must be abelian. In fact, if the order of such a subgroup Pm is />*", then each of its operators of order p must be invariant, since the group of isomorphisms of the cyclic group of order p is of order /> 1. Suppose that Pm contains a cyclic subgroup C^ of order P" such that Ca involves non-invariant operators. If this were possible Pm would contain an operator ^ which would transform C into itself without being commutative with every operator of Ca. As the operators of Ca would also trans- form into itself the cyclic group Sp generated by 5, it follows that the commutators, involving elements from Ca and Sp, would be in both of these cyclic groups. These commutators would therefore be in the central of the group generated by Ca and Sp. We may suppose that s was so selected that s^ is commutative with every operator of Ca- As the group gener- ated by Ca and this 5 would contain non-invariant operators which would not generate the commutator subgroup of order p, it results that Pm must be abelian. We have now proved that every Hamilton group is the direct product of a Hamilton group of order 2"* and some abeUan group of odd order. Hence it remains only to determine the possible Hamilton groups of order 2*". As an instance of such a group we may cite the quaternion group. We shall first prove that such a group H cannot involve any operator whose order ex- ceeds 4. In fact, if H contained an operator s whose order exceeds 4, we could find a subgroup in H, by the method used above, which would contain a non-invariant operator which would not generate the commutator of this subgroup. Hence we can assume that the order of every operator of H is either 2 or 4. Moreover, the operators of order 2 are contained in the central of H, while each operator of order 4 is transformed either into itself or into its inverse by every operator of H. Two of these non-commutative operators of order 4 must have a com- mon square, and hence any two such operators must generate the quaternion group. 47) HAMILTON GROUPS 117 The group generated by any one such quaternion group and the operators of order 2 contained in II must coincide with //. In fact, if an operator s oi H were not contained in this group this operator and this group would generate a group which would involve more operators of order 2. As this is contrary to the h>T)othesis, we have established the following theorem: Every possible Hamilton group is the direct product of a quaternion group, an ahelian group of order 2"* and of type (1, 1, 1, . . .), and an abeliati group of odd order * A group which is a direct product of two groups is some- times called a divisible group. If it is not a direct product it is said to be indivisible. Hence the quaternion group is the only indivisible Hamiltonian group. The only indivisible abelian groups are the cyclic groups whose orders are powers of prime numbers. If an abelian group is written as the product of indi- visible groups, the orders of these groups constitute the largest possible set of invariants of the abelian group. EXERCISES 1. The commutator quotient group of a Hamilton group of order 2* is abelian and of type (1, 1,1,...). 2. The number of the possible Hamilton groups of order 2";fe, k being any odd number, is equal to the number of the abelian groups of order k. 3. Every Hamilton group has the four-group as a group of inner iso- morphisms. 4. Two arid only two of the operators of a Hamilton group are character- istic. 5. Let g=Pi"Pt" . . . ^x"'^, where pu Pt, . . ,P\ are distinct primes. Necessary and sufficient conditions that all existing groups of order g shall be abelian are: (1) each otj^2; (2) no p"J l is divisible by one of the primes />i, p>, . . , P\. Cf. L. E. Dickson, Transactions of the American Mathematical Society, vol. 6 (1905), p. 201. * This theorem, together with various other theorems relating to Hamilton groups, was proved by G. A. Miller, Bulletin of the American Mathematical Society, vol. 4 (1898), p. 510. Some cf these theorems were proved several years later by E. Wendt, Mathematische Annalen, vol. 59 (1904), p. 187. In the fol- lowing volume of this journal Wendt corrected this oversight. CHAPTER V GROUPS WHOSE ORDERS ARE POWERS OF PRIME NUMBERS 48. Introduction. It has been observed in 11 that if />" is the highest power of the prime p which divides the order of a group G then G must involve at least one subgroup of order p^, and if G involves more than one such subgroup, all the sub- groups of this order (Sylow subgroups) form a complete set of conjugates. These facts indicate that it is especially important to know the fundamental properties of Syloiv's groups; * that is, of groups whose orders are powers of prime numbers. For- tunately all these Sylow groups have unusually interesting properties in common and they offer more easy avenues of penetration than the groups whose orders are arbitrary num- bers. A strong instrument of attack here, as well as in many other places in group theory, is the concept of complete sets of con- jugates. Each non-invariant operator of a non-abelian group G of order p"* belongs to a complete set of p conjugates, since such an operator is transformed into itself by all the operators of a subgroup whose order is p^, /3<w. Hence all the non- invariant operators of G occur in sets, such that each set involves a power of p conjugate operators, and each non-invariant operator occurs in one and in only one set. The total number of the non-invariant operators must therefore be of the form pk] and, as there are />'" 1 operators besides the identity in G, there must be an invariant operator of order p in G. This The groups whose orders are powers of prime numbers are also known as primary groups. G. Frobenius and L. Stickelberger, Journal reine angav. Math., vol. 86 (1879), p. 219. They are sometimes called prime-power groups. In view of the unusually large number of useful theorems in this field these groups have been said to constitute the El Dorado of the theory of groups. Bulletin oj the American Mathemalical Society, vol. 6 (1900), p. 393. 118 48] INTRODUCTION TO PRIME-POWER GROUPS 119 theorem was first proved by L, Sylow * and it may be regarded as the most important theorem relating to the prime-power groups. It has been observed that the totality of the invariant operators of any non-abelian group constitutes an important subgroup known as the central. With respect to its central, G is isomorphic to a group of order />"', m' <m. This quotient group must also have a central subgroup, if it is non-abelian, and this gives rise to a second quotient group of order />"", m" <m' . By continuing this process we must arrive at an abehan quotient group. It is a matter of considerable impor- tance to observe that this abelian group is never cyclic. In fact, this is a special case of the theorem proved in 28 that the central quotient group of a non-abeUan group is always non-cyclic. The subgroup of G which corresponds to an invariant sub- group of order p in the central quotient group of G is abelian, but includes operators which are not in the central of G. Hence it results that eroery non-abelian group of order p^ contains an invariant abelian subgroup whose operators are not separately invariant under the group. Since we can always arrive at the identity by forming successive central quotient groups of G it results that G must have at least one invariant subgroup whose order is an arbi- trary divisor of the order of G. Suppose that ^i, ^2, , Hp represent any complete set of conjugate subgroups of G. Since each of these subgroups is transformed into itself by a subgroup of G whose order is a power of />, it results that p is also a power of p. Hence each of these H's must transform into itself each one of at least /> l of the other ^'s, since it transforms itself into itself, and since it must transform a multiple of p of these conjugates among themselves. That is, each one of a complete set oj conjugate subgroups of a group of order p^ is transformed into itself by at least p \ others of the set, if the set includes more than one subgroup. In particular, every subgroup of order />""' in a group of order />" is invariant under * Malhcmaiische Annalen, vol. .'> (1872), p. 584. i20 CRIME-POWER GROUPS [Ch. V this group. As has been observed in 28, this theorem is also a special case of the following theorem: If Hi and H2 are two conjugate subgroups of G, then the index of Hi or H2 under G is greater than the index under Hi or H2 of the cross-cut of Hi and H2. 49. Invariant Abelian Subgroups. From the fact that every group of order />" contains at least p invariant operators and that its central quotient group has the same property, it results that every group of order p"*, m>2, contains a subgroup of order p^~^ which involves p^ invariant operators. In a similar way we observe that every group of order p^, m>5, contains a subgroup of order p^-'^-^ which involves p^ invariant opera- tors. In general, every group of order />*", w>(a+2)(a 1)/2, contains a subgroup of order ^-1-2-3-,.. -(a-l) _ j,m-o(a-l)/a which involves />" invariant operators. As the group formed by these invariant operators corresponds to an invariant sub- group in the quotient group, it results that it is invariant under the . entire group. Since every invariant subgroup of order pf* in a group of order p^ is contained in an invariant subgroup of order p"'^^, it results from the above that every group of order />", m>a{a+l)/2, contains an invariant abelian group of order //*+*. In other words, every group of order />"*, w>/3(/3 1)/2 contains an invariant abelian subgroup of order p^. In the special case when p = 2 this theorem may be expressed in a little more general form as follows: Every group of order 2*", m f i3(i3- 1)/2, /3> 3, contains an abe- lian subgroup of order 2^ *. The proof of this extended theorem is short if the preceding developments are employed. In fact, it has been proved that G involves a subgroup of order pm-l-2- ... -(/3-3) _ Am-(/3-2)J-3)/2 /3>3 * In fact, there is always an invariant abelian subgroup of order 2^ when the given conditions are satisfied. Cf. Messenger of Mathematics, vol. 41 (1912), p. 28. 491 INVARIANT ABELIAN SUBGK0UP8 121 whose centralis of order at least />^"*, whenever m^(0\) (/3- 2)/2. When w = /3(/3- 1)/2 the order of this subgroup is pfi(fi-l)/2-l0-2)(0-Z)/2 _ f,20-3 The quotient group with respect to the given central is of order p^~^. If this quotient group contains operators of order ^, G must evidently involve an abelian group of order p^. It remains therefore to consider the case when this quotient group does not involve any operator of order p^. li p = 2 we may assume that this quotient group is abelian, and hence we shall confine our attention, in what follows, to this special case. We are thus led to consider the possibility of constructing a group K of order 2^'^, having a central C of order 2^"^ which leads to an abelian quotient group of t>pe (1, 1, 1, ...). If we arrive at a contradiction by assuming that K does not include an abeUan subgroup of order 2^ our theorem is proved. If K existed, all the operators of C besides the identity would be of order 2, since all of these operators would be commutators of K. Moreover, each of the .non-invariant operators of K would be transformed under K into itself multiplied by all the operators of C. Let Ki represent any subgroup of order 2^""* and involving C Each of the non-invariant operators of Ki is transformed under Ki into itself multiplied by all the operators of a subgroup of order 2^"^ contained in C. The multiplying subgroups for two distinct operators (mod C) of Ki must be distinct, other- wise the operators of the group of order 4 (mod C) generated by these two operators would have to be transformed, by an operator of K which is not also in Ki, into themselves multi-' plied by the operators of a group of order 4 which has only the identity in common with the given subgroup of order 2^"' in C. As this is clearly impossible it results that all the dif- ferent non-invariant operators of Ki are transformed under Ki into themselves multiphed by all the different subgroups of order 2'*-' in C. From the preceding paragraph it results that there is a (1, 1) correspondence between the operators of A'l and the subgroups 122 PRIME-POWER GROUPS (Ch. V of order 2^"' in C such that each operator of K\ is transformed under K\ into itself multiplied by the various operators of the corresponding subgroup. Let h be any non-invariant operator of A'l and consider all the possible subgroups of order 4 in the quotient group of K\ with respect to C, such that each of these subgroups involves the operator corresponding to /i. Any oper- ator (p) of K which is not also in K\ transforms each of these subgroups into itself multiplied into a subgroup of order 4 con- tained in C. Let/i, ^2, . . , //3-2 represent a set of operators of K\ which correspond to a set of independent generating operators in the given quotient group and assume that ti~^S2h=S\t2, h-'^Szh=S2h, , /i"^-y^-2^i=-J/s-3^/j-2- The subgroup {h, ti) is transformed by p into itself multiplied by a group of order 4 which does not involve s\. In general, the subgroup (ii, /), a = 2, 3, . . . , jS 2, is transformed by p into itself multipHed by a subgroup of order 4 of C which does not involve 5_i, and {h, ta^tat ta^ is transformed by p into itself multiplied by a subgroup of order 4 which does not involve 5a._i5a,_i . . . 5j^_i, ai, 0:2, . . . ,ax = \,2, . . . , /3 2. As p must transform h into itself multiplied by an oper- ator which is common to all of these subgroups of order 4 and as 5i, 52, . . . , Sp-z are independent generators of a group of order 2^"^, it results that pti=tip, which is contrary to the hy- pothesis. That is, we have arrived at a contradiction by assum- ing that K does not involve an abeUan subgroup of order 2^ and hence the theorem under consideration has been proved. EXERCISES 1. In a group of order ^"' the order of the commutator subgroup cannot be greater than p"*~^. 2. In a non-abelian group of order />' each of the non-invariant operators belongs to a complete system of p conjugates. 3. There is one and only one non-abelian group of order />', p>2, which is conformal with the abelian group of type (1, 1, 1). 4. If a non-abelian group of order />*", P>2, contains an operator of order p^~ * its commutator subgroup is of order p and there is only one such group. 501 NUMBER OF THEIR SUBGROUPS 123 5. Every non-abelian group of order ^"* contains an invariant conimuta- tor of order p. 6. If a group of order 3"* contains no operator of order 9 all of its opera- tors in any complete set of conjugates are commutative.* Suggestion: If 5i, Si are any two operators of such a group it results that {sis^^= Si- SiSiSi^ Si'^S\Si= {siSi}y=Si-SiHxSi-StSiS%^= 1. 7. A necessary and sufficient condition that a group of order ^"* is abe- lian is that more than />"*"* of its operators corresponds to their inverses in some automorphism of the group. 8. If two non-commutative operators of a group of order />"*, P>2, correspond to their inverses in an automorphism of the group their com- mutator cannot correspond to its inverse in this automorphism. 50. Number of Subgroups in a Group of Order p^. We shall first determine the form of the number of subgroups of order />"*"^ in a group G of order f^. Any two subgroups of order f^~^ must have p^~'^ operators in common, and these common operators constitute a group which is invariant under G. They must therefore include all the commutators of G and also the pth. powers of every operator of G. If H is composed of all the operators which are common to all the subgroups of order />"*"^ contained in G it must include all the commutators of G as well as the p\h powers of all its operators. From this it results that the quotient group corresponding to H is abelian and of type (1, 1, 1, ...). Each subgroup of order />"*"* in G must correspond to a subgroup of index p in this quotient group. In Chapter IV, 40, we proved that the number of these subgroups of index p is equal to the number of the subgroups of order p in this quotient group. Hence the theorem: The number of subgroups of order />""* contained in a group G of order />"* is always of the form p^-l p-\ To obtain the exact number of these subgroups it is neces- sary to observe that p*" is the order of the quotient group of G with respect to the group formed by all of its operators W. Bumside, Quarterly Journal of Mathematics, vol. 33 (1901), p. 231. 124 pr;me-power groups ICh. V which are common to all of its subgroups of order T''- I" the special case when G is abelian and of type (1, 1, 1, ... ), X = w; but in all other cases, m>\. Since in any group every subgroup which involves half the operators is invariant, it results from the above that the number of subgroups whose order is one-haK the order of the group is always of the form 2_i. In particular, there is no group which involves exactly 5 or 9 subgroups of half its order. The last two statements are evidently not restricted to groups whose orders are powers of a single prime. In fact, it results from the given method of proof that the number of the invariant subgroups of index * in any group whatever which contains at least one such invariant subgroup is of the form {p^-l)/{p-l)- This is known as Bauer's theorem* Having found the number of the subgroups of order p"^ in G, we shall now consider the other extreme case and deter- mine some property of the number of its subgroups of order p. The invariant subgroups of order p which are contained in G clearly generate a group of order / which^contains (^pP _ !)/(/, - 1) distinct subgroups of order p, where /3 > 1 . The non-invariant subgroups of order p may be divided into complete sets of conjugates, each set containing p'', 7>0, distinct groups. Hence the total number of subgroups of order p in any group whose order is a power of p is always of the form l-\-kp. From this theorem it results immediately that the number of the sub- groups of order />"+' which contain a given subgroup of order f* is always of the form l-\-kp. We proceed now to consider the number of subgroups of order /> in G, where p is an arbitrary divisor of />. We assume at first that a<m, and denote by r the number of subgroups of order p" in G and by r+i the number of these subgroups of order />"+^ Let Sx represent the number of the subgroups of order />-+' in which a given subgroup of order p" occurs, and let Sy denote the number of subgroups of order p' contained in a given subgroup of order r""'- We then count each sub- * H. Hilton, Introduction to the Theory of Groups of Finite Order, Oxford, 1908, p. 145. 5 501 NUMBER OF THEIR SUBGROUPS 125 group of order r-. as many times as it contains different subgroups of order ^ and tlius arrive at tl,e following equaUon: I-l 1,-1 As both . and .are congruent to unity modulo p it results that r =r^,(mod p). As n^l(mod p) it results that r.^1 (mod p) and hence tite number of the subgroups of order p" in a group of order />" is always of the form 1 -{-kp. It is now very easy to prove that the number of the sub- groups of order r in any group G' whose order is divisible by P IS always of the form l^kp, even if the order of G' is not a power of p. If a subgroup of order ^ in G' is not invariant under some one of the Sylow subgroups of order r in G' it evidently belongs to a complete set of ps conjugates. Hence we niay confine ourselves to these subgroups of order ^ in G wbch are mvariant under a particular subgroup of order r m G All of these must occur in this Sylow subgroup of order r and hence their number is of the form 1 ^kp. This proves tiiat the total number of the subgroups of order /,- must also be of this form, so that we have proved the following theorem due to Frobenius: The total number of the subgroups of order p- in any .roup whose order is divisible by p- is of the form 1+kp -^ ^ This theorem holds whether the order of the group is or is not a power of a single prime, and it may be regarded as an exten^n of Sylow 's theorem. It should, however,\e obse'ed that the subgroups of order p" are not always conjugate when they are not Sylow subgroups. . ^ J h ^ wnen If the group G of order ^ contains at least one abelian subgroup of order p^ it is easy to prove that the number of Jts abehan subgroups of order /.^ is of the form 1 -\-kp. We proceed to prove this theorem. If G contains an abelian subgroup B of order r the totahty of the operators of G which are com mutative with every operator of 77 forms a group H' which mcludes aU the abelian groups of order p"^^ that are con tamed m G and include H. Hence the abelian subgroups of 126 PRIME-POWER GROUPS (Ch. V order />*"*"* in G which include H correspond to the subgroups of order p in the quotient group of H' with respect to E. As the number of these subgroups of order p is of the form \-\-kp whenever H and H' are not identical, we have proved that an abeUan group of order p" contained in G is found in l-\-kp abeUan subgroups of order /?""*" ^ whenever it is found in at least one abelian subgroup of this order. Suppose that G contains at least one abelian subgroup of order /j""^^ and let Ta+i represent the number of its abelian subgroups of this order, while r represents the number of the abelian subgroups of order p" contained in G and included in abelian subgroups of order p"^^. By counting each abelian subgroup of order />""" ^ as many times as it contains subgroups of order />" and denoting by Sx and Sy respectively the number of the abelian subgroups of order ^"+^ in which a given subgroup of order p" occurs, and the number of subgroups of order />" in a given abelian subgroup of order />"*" ^ we obtain, as before, the equation S 5^= 2 Sy. x-i v-y Since both Sx and ^i, are congruent to unity modulo p we have proved that ra = ra+i (mod p). As ri = l (mod p) and r2^1 (mod p) whenever G contains an abelian subgroup of order p'\ there results the theorem : The number of the abelian subgroups of order p^ in any group of order p^ is either zero or of the form l-{-kp. It should not be inferred that the number' of the abelian subgroups of a group of order p^ is always either 0, or of the form 1 -\-kp. The number of the abelian subgroups of index p in a non-abelian group of order p^ is either 1 or 1 -\-p, if this number exceeds zero, according to example 3 of the following exercises; but there are non-abelian groups of order />" which contain exactly two abelian subgroups of index p-. The pos- sible numbers of abeUan subgroups in a non-abelian group of order />" have not yet been determined. By means of Bauer's theorem and the properties of the ^subgroups it is easy to prove a fundamental theorem relating 501 NUMBER OF THEIR SUBGROUPS 127 to the various possible sets of independent generators of any prime power group. As every maximal subgroup of a group of order f' is of order p^-\ it results directly from the defini- tion, that the 0-subgroup of any group of order f is the cross- cut of all its subgroups of index p. Hence the (^-subgroup of a group of order f" can also be defined as its smallest invariant subgroup which gives rise to an abelian quotient group of type (1, 1, 1, . . . ). If the order of this quotient group is p", it follows that a is the number of independent generators in every possible set of such generators of this group of order />*". In particular, every possible set of independent generators of any prime power group involves the same number of operators. That is, the number of operators in each of the possible sets of indepen- dent generators of a Sylow group is an invariant of this group. From the preceding paragraph it results directly that a necessary and sufficient condition that the <^-subgroup of a given group of order p^ be the identity is that this group be the . abeUan group of type (1, 1, 1, . . . ). Hence there is one and only one group of order />", p being any prime number and m being any positive integer, which has the identity for its 0-subgroup. The number of operators in every set of inde- pendent generators of this group is m. In every other group of order />" the number of these independent generators is less than w, and there is at least one group of order f in which this number is any arbitrary positive integer from 1 to m. EXERCISES 1. The number of abelian subgroups of order /> in any group is either or of the form 1 -{-kp, even if the order of the group is not a power of p. 2. If an abelian group of order p"^ has p'- for one invariant while all of its other invariants are equal to p, the number of its subgroups of order ^-1 is P^l^, p-\ 3. A non-abelian group of order p''^^ contains 0, 1, or p + l abelian sub- groups of order />*; in the last case it contains />'"' invariant operators. 4. If a group of order 2* contains only one subgroup of order 2 it is either cyclic or dicyclic, and if a group of order />"*, p>2, contains only one subgroup of order p it must be cyclic* W. Bumside, Theory of Croups, 1911, p. 132. 128 PRIME-POWEll GROUPS (Ch. V 5. If a group G of order />"* contain only one subgroup of order p", 1 <a<m, the group is cycltc. Suggesliott: Every operator of G which is not in the subgroup of order p" must generate this subgroup, since every subgroup of order p" in a group of order />"* is contained in a subgroup of order />"*'*. 6. The number of the invariant subgroups of order p" in any group of order p'^ is of the form 1 +kp whenever w > . 7. There are just four non-abelian groups of order 2" involving a cyclic group of order 2*""*, m>S. 8. Prove that the number of the subgroups of order p in any group of order p" is of the form 1 +kp, by observing that the number of operators of order p" in such a group is of the form kap"~^{p 1), so that ^m_i^"2\/-^(/.-l).. 9. Find a group of order 18 which contains only two invariant sub- groups of order 3, and hence prove that the number of invariant subgroups of order />" in a group need not be either or of the form I -{-kp. 51. Number of Non-cyclic Subgroups in a Group of Order J*"*> i^>2. It has been observed that the total number of the subgroups of order />** in any group G of order p^ is of the form \-\-kp whenever m>a, and hence the number of the invariant subgroups is always of this form. The number of the abelian subgroups is again of this form whenever a < 4 and this number is greater than zero. We proceed to prove that the number of the non-cyclic subgroups is always of the form l-\-kp when- ever p>2. This proof can easily be effected by a method employed above. It is easy to prove that the number of the non-cyclic subgroups of order ^" in a group of order p"'^^ is of the form l-\-kp whenever this number is not zero and p>2 by showing that there must be p cyclic subgroups of order />* whenever there is one such subgroup. The number of the subgroups of order />""*" S which contain a given non-cyclic group of order p and are themselves contained in a group of order p^, is also of the form l+kp. Let fa and r^+i represent respectively the numbers of the non-cyclic subgroups of order p and />"*"', and let Sx represent the number of the subgroups of order />"""' in which a given non-cyclic subgroup of order p occurs while Sy denotes the num- 521 NUMBER OF NON-CYCLIC SUBGROUPS 129 ber of non-cyclic subgroups of order p" contained in a given subgroup of order p""^^. We then count each subgroup of order />+^ as many times as it contains a non-cyclic subgroup of order />" and thus arrive at the equation 2j Sx ^ ^ Sy% X-l y.l Since both Sx and Sy are congruent to unity modulo p, it results that fa^Ta+i (mod p) whenever G contains at least one non- cyclic group of order p" and m>a. Since every non-cyclic group of order p^ contains at least one invariant non-cyclic group of order p^ whenever p>2, there results the theorem: The number of the non-cyclic subgroups of order p" in any non-cyclic group of order p^ is of the form 1+kp whenever l<a<m andp>2.* 52. Number of Non-cyclic Subgroups in a Group of Order 2"*. It is known that there are three non-cycUc groups of order 2"* which contain a single cyclic subgroup of order 2', 2<a<m; viz., the three groups of order 2"* which involve a cyclic subgroup of order 2*""^ and transform a generator of this subgroup into its inverse or into its (2"*~^ l)th power. It is not difficult to prove that these are the only possible non-cyclic groups of order :'^ that contain an odd number of cyclic subgroups of order 2". If a non-cyclic group G of order 2*" contains an odd num- ber of cyclic subgroups of order 2", at least one of these is in- variant under G. If this were the only cyclic subgroup of order 2 in G there could not occur in G two cychc subgroups of order 2'^^, /3>0, since one of these would be transformed into itself by one of its conjugates, and hence one of these and some operator in a conjugate subgroup would generate a group of order 2^+^+^ which would be conformal with the abelian group of t3^e {a-i-fi, 1). As this group would involve two cyclic groups of order 2", it has been proved that if G contains only one cyclic subgroup of order 2", it contains no more than one cyclic sub- group of order 2"+^, a-{-^<m, /3f 0. * Proceedings of the London Mathematical Society, vol. 2 (1904), p. 142. 180 PRIME-POWER GROUPS [Ch. V Suppose that G contains one and only one cyclic subgroup of order 2** and that its largest cyclic subgroup H' is of order 2". Since every subgroup of order 4 in the group of isomorph- isms of W involves the operator of order 2 which is com- mutative with just half of the operators of H^ it results that ai=m 1, otherwise G would involve a subgroup which would be conformal with the abelian group of type (ai, 1). From the known properties of the groups of order 2" which involve operators of order 2*""^ and the result just obtained, it follows that G must be one of the three groups mentioned above whenever it contains one and only one cycUc subgroup of order 2. It remains to prove that G can involve only one such subgroup whenever it involves an odd number of cyclic sub- groups of this order. Suppose that G contains 2n+l cyclic subgroups of order 2**. At least one of them, Hi, is invariant under G. Let H2 repre- sent any other and let H'2 be a subgroup of H2 such that the group (^1, H'2) is of order 2"+^ Hence {Hi, H2') has just two cyclic subgroups of order 2" and at least 2**"^ invariant operators. These two cyclic subgroups have just 2""^ common operators and we proceed to prove that there is an even num- ber of cyclic subgroups of order 2 in G such that they all have 2"~* operators in common. If there is in G another cyclic subgroup of order 2** which has exactly 2"^ operators in common with H\, it is trans- formed either into itself by (^1, H'2) or it is transformed into a power of two distinct conjugates such that all of these have 2**"^ operators in common with H\. In the former case its generators are either invariant under all the operators of order 2" in (^1, H'2) or they are transformed by at least some of these operators into themselves multiplied by the operators of order 2 in Hi, while the remaining operators of {Hi, H'2) transform each of these generators into themselves. In this case (^1, H'2) is therefore invariant under these subgroups, that is, {Hi, H'2) is invariant under all the subgroups of order 2", contained in G, which have 2""^ operators in common with Hi and are transformed into themselves by (Hi, H'2). 52l NUMBER OF NON-CYCLIC SUBGROUPS 131 If Hz is such a subgroup it is clear that (//i, '2^ B3) is conformal with an abelian group and hence involves an even number of cyclic subgroups of order 2". We consider now all the other cyclic subgroups of order 2 contained in G, which have 2"~* operators in common with Hi and are transformed into themselves by (Hi, H'2, Ha)- If such a subgroup 7/4 exists it is clear that (Hi, H'2, H3, H4) must again be conformal with an abelian group and hence it involves an even number of cyclic subgroups of order 2". By continuing this process it results that an even number of cyclic subgroups of order 2**, contained in G, have 2"~^ operators in common with Hi. All of these cyclic subgroups of order 2" form a set. If G contains a cyclic subgroup of order 2" which is not in this set, this subgroup must belong to another set which involves an even number of distinct cycb'c subgroups of order 2"; that is, these cyclic subgroups of order 2" are found in distinct sets such that no two have a subgroup in common and such that each set involves an even number of cyclic subgroups. This completes the proof of the theorem: if a group of order 2"* involves an odd number of cyclic subgroups of order 2, a>2, this number is unity. Combining this with the theorem proved above, it results that if we include the cyclic group there are just four groups of order 2*" such that each involves an odd number of cyclic subgroups of order 2", a>2.* In each of these groups there is only one such subgroup. The four groups mentioned in the last theorem contain one, 2w-2_j_j Qj. 2-3-j-i cyclic subgroups of order 4. We proceed to prove that these are the only groups of order 2" which con- tain an odd number of cyclic subgroups of order 4. The method of proof is similar to the one employed above. Let G be any group of order 2" which contains an odd num- ber of cyclic subgroups of order 4. At least one of these sub- groups (Ki) is invariant under G, and at least half the oper- ators of G are conmiutative with a generator (s) of Ki. It will be proved that the subgroup Gi formed by these 2*"' operators must be cyclic. If Gi were not cyclic, A'l would be * Transactions of the American Mathematical Society, vol. 6 (1905), p. 58. 132 PRIME-POWER GROUPS [Ch. V contained in an abclian subgroup of type (2, 1). This sub- group would contain two cyclic subgroups Ki and K2 of order 4, having a common square. It will now be proved that G would then contain an even number of cyclic subgroups of order 4 having s^ in common. If this number were odd, (Ki, K2) would transform at least one of the remaining ones (/V3) into itself. The commutator subgroup of {Ki, K2, Kz) is generated by s^ and hence this group contains an even number of cyclic subgroups of order 4 and all of these subgroups contain s"^. Hence there would be another invariant cyclic subgroup K^ involving s^. The number of cyclic subgroups of order 4 in {Ki, K2, K3, K4) is again even, since the commutator subgroup is still generated by S' and all of these subgroups include s^. This follows almost directly from the fact that the product of an operator of order 4 in (Ki, K2, K3, K4) into an operator of order 2 is of order 4 when the two factors are commutative, and of order 2 when they are not commutative, while the converse is true if the second operator is of order 4. As this process could be continued indefinitely if there were an odd number of cyclic subgroups of order 4 which contained s^, it results that the number of these subgroups is even. If there were an odd number of cyclic subgroups of order 4 in G, which did not contain s^, one of these and s^ would again generate the abelian group of type (2, 1), and the number of those involving the same subgroup of order 2 would again be even. As similar remarks would apply to all the other possible cyclic subgroups of order 4 we have proved that Gi is cyclic whenever G contains an odd number of cyclic subgroups of order 4. This proves the statement in the first paragraph of this section, since the other non-cyclic group of order 2*" which contains a cyclic subgroup of order 2*""^ contains an even number of cyclic subgroups of order 4. Combining the results of this section we conclude that every group of order 2", w>3, which contains an odd number of cyclic subgroups of order 4, contains just one cyclic subgroup of order 2", where a can have any value from 3 to m 1; and 52] NUMBER OF NON-CYCLIC SUBGROUPS 133 every group which contains only one cyclic subgroup of order 2" contains an odd number of cyclic subgroups of order 4. For each value of a and m there are three such groups; hence there is an infinite system of groups of order 2*" which contain an odd number of cyclic subgroups of composite order. When w = 3 there are only two groups having this property, viz., the quaternion group and the group of movements of the square. If all the non-cyclic groups of order p^{m>Z, p an arbitrary prime) were determined, there would be just three among them in which the number of cyclic subgroups of composite order would not always be a multiple of p. In these three special cases the number of cycHc subgroups of every composite order is not divisible by p. In the exceptional groups noted above the number of the subgroups of order 2 is =1 (mod 4). That this number is =3 (mod 4) in every other non-cyclic group of order 2"* is a direct consequence of the fact that the number of cych'c sub- groups of order 4 in all these groups is even. From this fact it results that the number of operators whose order exceeds 2 is divisible by 4, since every cyclic subgroup of order 2' con- tains 2'~^ operators which are not found in any other subgroup whose order is = 2'. Hence the given system of groups is composed of all the groups of order p*^ in which the number of subgroups of order p is not =l-\-p (mod p^). That is, the groups of order /?*" in which the number of cyclic subgro\ips of composite order is not divisible by p coincide with those in which the number of subgroups of order p is not of the form l-\-p-\-kp^. EXERCISES 1. The number of subgroups of order p in any non-cyclic group of order /", p>2, is of the form l+p+kp\ Suggestion: Consider the forms of the number of the operators of orders p\p', etc. 2. The number of cyclic groups of order p^,p>2 and /3> 1 , is of the form kp whenever the Sylow subgroups of order />"* are non-cyclic. 3. If a group G of order p"* contains exactly p cyclic subgroups of order p", these subgroups generate a characteristic subgroup of order p""*"^ under G, and this subgroup is either abelian and of type (a, I) or it is the non-abelian group which is conformal with this abelian group. 134 PKlME-ruWEli GROUPS [Ch. V 4. There are just three groups of order 3* each of which contains only three cyclic subgroups of order 9. 53. Some Properties of the Group of Isomorphisms of a Group of Order p^. If Ga is any invariant subgroup of G, then Ga contains at least one invariant subgroup of G whose order is an arbitrary divisor of the order of Ga- Let Go, Gi, G2, . . . , Gm be a series of invariant subgroups of G whose orders are respect- ively 1, P, p"^, ' ' , P^, so that Gm=G; and suppose that each of these subgroups except the last is included in the one which follows it. Let / be any operator whose order is a power of p in the group of isomorphisms of G. Such an operator is said to effect a p-isomorphism of G. Since / and G generate a group whose order is a power of p, we may suppose that / transforms all of the operators of each one of the given series of subgroups into themselves multiphed by those of the preceding subgroup. That is, the commutator of t and any operator of G^ is in G^-i, i3 = l, 2, . . . ,w. We proceed to prove that the order of / cannot exceed p*^~^. In fact, if Ifi is any operator of Gp in the above series, it results from the given conditions that Hence /***" must be commutative with every operator of G, and, as t is an operator in the group of isomorphisms of G, it results from this that the order of / is a divisor of p^~^. In other words, the p^'^ power oj every operator whose order is a power of p in the group of isomorphisms of a group oJ order p^ is the identity. When G is the cycUc group of order p^, p>2, it is known that the group of isomorphisms of G involves oper- ators of order p^~^, viz., those operators which give rise to commutators of order pr^~^. We proceed to prove that the group of isomorphisms of a non-cyclic group of order />"*, p>2 and w>3, cannot involve any operator of order p^'"^. If G is such a group it follows from the preceding section that we may assume that G2, in the above S 631 THEIR GROUP OF ISOMORPHISMS 135 series of invariant subgroups, is non-cyclic. If / is an operator in the group of isomorphisms of G and if the order of / is a power of p, it results that /" is commutative with every operator of G2 and that it transforms the operators of G into themselves multiplied by operators in Gm-2. Similarly /"* is commutative with every operator in G3 and transforms every operator in Gm into itself multiplied by an operator in Gm-3- In general, /"* transforms every operator in G into itself multiplied by operators in Gn,-a-i- As t" transforms the operators of G into itself multiplied by operators in G2 and as these operators are com- mutative under this power of t, it results that 1-2 t'' =1. That is, the order of every operator in a Sylow subgroup of order p^ of the group of isomorphisms of a group of order />" is a divisor of />*"* whenever p> 2 and m> 3. The last theorem evidently also applies to all groups of order 2" which involve a non-cyclic invariant subgroup of order 4, and it is well known to be true as regards the cyclic group of order 2*". It is, however, not true as regards the dihedral or the dicyclic group of order 2", since there is evidently an opera- tor of order 2"*~^ in the group of isomorphisms of each of these groups; viz., the operator which is commutative with each operator of the cyclic subgroup of order 2"*"^ but transforms the non-invariant operators of orders 2 and 4 respectively into themselves multipUed by an operator of order 2**~^ These two infinite systems of groups therefore are instances of groups whose groups of isomorphisms contain operators of the largest possible order in accord with the theorem of the second para- graph of the present section. As the remaining group of order 2" which does not involve an invariant non-cyclic subgroup of order 4 involves 2"*"^ non- invariant cyclic subgroups of order 4, its group of isomorphisms may be represented as an intransitive substitution group on 2""^ letters. From this it follows directly that its group of isomorphisms cannot involve any operator of order 2""*, m>3. Hence we have proved the theorem: 136 PRIME-POWER GROUPS (\i. V The only groups of order />"*, m> 3, wJwse groups oj isomorph- isms involve operators of order p"*~^ are the cyclic group when P>2y and the dihedral and the dicyclic groups when p = 2. Every operator whose order is a power of /> in the group of isomorphisms of G generates with G a group whose order is a pKJwer of p. As G is an invariant subgroup of this group it results that this group involves a series of invariant subgroups Go, Gi, G2, . . . , Gm which are such that / transforms the operators of each of these subgroups into themselves multiplied by those of a preceding subgroup. Moreover, whenever / has this property its order is a power of p. Hence the existence of such a series of subgroups is a necessary and sufficient condition that an operator / in the group of isomorphisms of G has for its order a power of p. In other words, A necessary and sufficient condition that an operator t in the group of isomorphisms of a group G of order p"* has for its order a power of p is that t transform every operator in tJie series of sub- groups Go, Gi, G2, ' ' , Gm = G of orders I, p, p^, . . _. , />"* respectively, into itself multiplied by an operator in the preceding subgroup. Hence t^ transforms every operator of Gp into itself mul- tiplied by an operator inG^-a-i, where jS a 1 is to be replaced by whenever i3a+l. From this theorem it is easy to deduce the following useful corollary: The order of the commutator of two operators, neither of which is the identity, of a group G of order p^ is less than the order of the smallest invariant subgroup of G in which either of these two operators occurs. In particular, if one of these two operators is of order p and generates an invariant subgroup, it must be commutative with every operator of G. This special case may also be regarded as a special case of the theorem that every invariant subgroup, besides the identity, in a group of order p^ involves invariant operators of order p. 54J SYLOW SUBGROUPS OF ISOMORPHISMS 137 The group of isomorphisms of the group of order p is the cyclic group of order /> 1, and hence it does not involve any operator of order p. It is easy to prove that the group of iso- morphisms of every group G of order p^, m>\, involves oper- ators of order p. In fact, when G is non-abelian its group of cogredient isomorphisms involves such operators and when G is abelian we may select any subgroup of order p^'^, and multiply the operators of the corresponding quotient group by those of an invariant subgroup of order p which is contained in the given subgroup of order p^~^. It is evident that the latter isomorphisms are also possible even when the group of order p^ is non-abelian. Hence the group of order p is the only group of order />"* whose group of isomorphisms does not in- volve operators of order p. EXERCISES 1. If G is the abelian group of order />*" and of type (1, 1, . . . ), its group of isomorphisms cannot involve any operator whose order is a power of p and exceeds p^^^ when m is even or ^<'"+i)/2 when m is odd. Suggestion: When a power of / transforms G into itself multiplied only by operators which are commutative with t, the ^th power of this power of t will be the identity. 2. The group of isomorphisms of the dihedral group of order 2, >2, is the holomorph of the cyclic group of order n. Suggestion. The group of isomorphisms of this dihedral group may be represented as a transitive substitution group of degree , and it in- volves an invariant cyclic subgroup of order n composed of all its opera- tors which are commutative with every operator of the cyclic subgroup of order n. 3. Find the group of isomorphisms of the dicyclic group of order 2", n>3. 4. If a group of order 2"* contains exactly two cyclic subgroups of order 2^, but no cyclic subgroup of any higher order, then m/3-f2. 64. Maximal Order of a Sylow Subgroup in the Group of Isomorphism of a Group of Order p"*. We proceed to deter- mine the maximal order of a Sylow subgroup of order />* con- tained in the group of isomorphisms of a group of order p^. To do this we assume again that a set of invariant subgroups 138 PRLME-roWKR GROUPS (Ch. V of orders I, p, p^, - - , p^ respectively of such a group G is represented by the symbols Go, Gi, G2, I Gmi and that these invariant subgroups have the property that each one except the last is included in the one which follows it. The operators of G which are not also in Gm-i may be transformed into themselves multiplied by at most p^~^ different operators. In general, the operators of Ga which are not in Ga-i are transformed into themselves multiplied by at most p"'^ opera- tors. Hence the order of a Sylow subgroup of order p^ in the group of isomorphisms of G cannot exceed p.p2 . . ^ . , pm-l _pm(.m-l)/2^ It is evident that when G is abelian and of type (1, 1, . . . ), its group of isomorphisms involves Sylow subgroups of order /)", where n has the maximal value given above. This is also the case when G is abelian and of type (2, 1, 1, ... ). It is not difl&cult to prove that these are the only abelian groups whose groups of isomorphisms contain Sylow subgroups of the given maximal order.* 56. Construction of all the Possible Groups of Order p"". In view of the simple properties of the group of order p"* various efforts have been made to determine all these groups for given values of m. When m = l there is evidently only one such group, and when m = 2 there are two possible groups. Each of these groups is abelian. Moreover, it has been observed that for any value of m the number of abelian groups is equal to the number of different partitions of m as regards addition. Hence the only difficulty in the determination of all these groups for small values of m relates to the possible non-abelian groups. In case m<5 there is very little difficulty here, but for larger values of m this difficulty increases very rapidly with m. As an abstract group is simply isomorphic with one and only one regular substitution group, we can determine all the * Cf. G. A. Miller, Transactions of the American Malhematical Society, vol. 12 (1911) p. 396. 65) CONSTRUCTION OF GROUPS OF ORDER p" 139 possible groups of order />" by determining all the regular substitution groups of this order. We proceed to indicate a method for constructing all the possible regular groups of order p^ on condition that all those of order p^~^ are known. Since every regular group of order p^ contains an invariant intransitive subgroup of order />"*~^ which is formed by a simple isomorphism between p regular groups of this order, we may first determine all the possible regular groups of order />" which involve a given group B. of order p^~'^. Let the p transitive constituents of H he Hi, H2, . . . , B. A group G of order />*" which contains H is generated by H and some substitution h which permutes the p systems of intran- sitivity of H cyclically, has its />th power in H, and transforms E into itself. Let / be a substitution of order p which permutes the systems of E transitively and is commutative with every substitution of E. As it may be assumed that /i and t permute these systems in the same way, we conclude that ht-^ is a sub- stitution 5 which does not permute any of the p transitive con- stituents of E. That is, tx =st, where it may be assumed that t transforms the corresponding letters of each of the p systems among themselves. As / can readily be obtained from the p constituents belonging to Ei, E2, . . . , Ep oi any substi- tution in E it remains only to determine s = siS2 . . . Sg, where Si, S2, ' ' , Sp are the constituents of s belonging to Hi, E2, . . . , Ep respectively. As Si, S2, . . . , Sp transform each of the constituents Ei, E2, . . . , Ep into itself they must be found in the holomorphs of Ei, E21 . , Ep, respectively. We proceed to prove that 52, Sz, . . . , Sp may be assumed to be corresponding substitutions in the groups of isomorphisms of E21 Ez, . . . , Ep, respectively, while ^1 is one of the />"*~^ substitutions obtained by multiplying the substitution in the group of isomorphisms of Ei which corresponds to ^2, 53, . . . , Sp by the f^~^ substitutions which are commutative with El and involve only letters of Ei. This theorem follows almost directly from the fact that the conjoint of a regular group has the same group of isomorphisms as the regular group itself, since both are invariant subgroups of the holomorph of 140 PRIME-POWER GROUPS [Ch. V this regular group, and the group of isomorphisms of a regular group is the subgroup composed of all the substitutions which omit a given letter in its holomorph. Hence we may reduce each of the substitutions which transform // in a certain way to one of the given form by transforming by a substitution in the direct product of the conjoints of H\, H2, . . , Up. From the preceding paragraph it results that the number of groups of order p"* which contain a given group 77 of order pm-i cannot exceed the order of the product of H and the order of its group of isomorphisms. Hence the number of the groups of order p'^ is always finite when p is finite. We shall be able, however, to reduce this apparently possible number very greatly. In the first place it should be observed that if 5/ is so selected that Si'soSs ... 5p is commutative with /, while si involves only letters in Hi, then si=si'^si where 5i" is commutative- with si'; otherwise (51^2 . . . Spty would not be in H. This restricts the number of choices of si" to the number of operators in H which are both in its conjoint and also inviariant under si. Another important restriction is that we evidently need to consider only one of the first p l powers of 5/5253 . . . Sp, since / may be transformed into any power without affecting 5i52 ... 5p by permuting the constituents H2, Hs, . . . , Hp cycUcally. When H is abelian, all the groups obtained by replacing 5/' by any of its powers prime to its order are conjugate, since the operators in the group of isomorphisms of an abelian group which transform each operator of this group into the same power are invariant under this group of iso- morphisms. To illustrate the method outlined above we proceed to determine the possible non-abelian groups of order p^, p>2. It has been observed that each of these groups involves a non- cyclic group of order p^, and hence we may use this for H. As the group of isomorphisms of // involves only one set of con- jugate subgroups of order />, the two substitutions 5/5253 . . . Sp and t may be supposed to be the same for all these groups. Moreover, 5/' is restricted to a single subgroup, and hence it is necessary to consider only two cases, viz., the case when 551 CONSTRUCTION OF GROUPS OF ORDER p* 141 Si" is the identity and the case when its order is p. That is, there cannot be more than two non-ahelian groups of order p^, p>2, in accord with the general theory which precedes this paragraph. From the present paragraph it results that one of these groups involves operators of order p^ while the other does not have this property. Hence there are two and only two non-abelian groups of order p^, p>2. It was proved in 33 that there are also only two such groups when p = 2. Among the groups of order p" which involve H the one gen- erated by H and / is of especial interest in view of its simple structure. In fact, it is abstractly the direct product of H and the group of order />. If we transform this group by ri^-^nu^ . . . r/-^ where ri is any invariant substitution of Hi while r^, a = 3, . . . , p, is the transform of ri with respect to t", there results a group generated by H and ri~^t. From this and the preceding theory it results that we need to use only two values for si" when n is cycUc and the entire group is abelian, as results also directly from the theory of the abelian groups. This general method can easily be extended so as to reduce the amount of labor necessary to determine all the possible groups of order f^ which involve a given subgroup of order p""'^ *; but the pre- ceding developments may suffice to point the way towards a penetration into this difficult subject. From the given illus- tration it results that we do not always need to use all the pos- sible groups of order />"*~^ for // in order to determine all the groups of order p^. In fact, when w = 4 we need to consider only the abelian groups of order p^ for H, since every group of order p* contains an abelian subgroup of order p^. EXERCISES 1. Determine the fourteen possible groups of order 16. 2. Determine the non-abelian groups of order />*, />>2, which involve no operator of order />-. Show that Si"= 1 in these cases and that there are two such groups when />>3, one having a commutator subgroup of * Aitkirican Journal of Mathematics, vol. 24 (1902), p. 395. 142 PRIME-POWER GROUPS [Cn. V order p and the other having such a subgroup of order p^. When ^=3 there is only one such group. 3. There are four distinct non-abelian groups of order />*, />>3, which involve an abelian group of order />*, but do not contain any operator of order ^*. When /=3 there are only two such groups. 4. The number m can be so chosen that the number of the distinct groups of order />'", />>2, which do not involve any operator of order p^ is greater than any given number. 5. Every intransitive Sylow subgroup of a symmetric group is the direct product of its transitive constituents, and each of these transitive constituents has a central of prime order. 6. If the degree of a symmetric group is =i&i^**+;fe2/'*~^+ . . +*a+i; *i' h, . . . , ka+\ being positive integers less than p, then the central of its Sylow subgroup of order p^ is of order />*+*+ +*. CHAPTER VI GROUPS HAVING SIMPLE ABSTRACT DEFINITIONS 56. Groups Generated by Two Operators Having a Common Square. If si, S2 represent two operators of order 2 they evi- dently satisfy the equation Si^ = S2^, and the cyclic group (S1S2) generated by S1S2 is invariant under ^i and 52. In fact, S1S2 is transformed into its inverse by each of the two operators Si, 52, and hence (^i, 52) is the dihedral group whose order is the double of the order of 51^2, as has been observed in 26. Hence the equations 51^=52^ = 1, (5i52)=l serve as a complete definition of the dihedral group of order 2, if we assume that the order of 51^2 is exactly n. Through- out the present chapter it will be assumed, unless the contrary is stated, that the condition 51"= 1 implies that the order of Si is exactly n. This fact is sometimes expressed by saying that si fulfils the condition si* = l, while the statement 5i satisfies the condition 5i" = 1 may imply merely that the order of Si is a divisor of n* We shall employ the terms fulfil and satisfy with these meanings throughout the present chapter, so that the equation 51" = 1 implies that si fulfils this condition unless the contrary is stated. Ji Si, S2 are any two operators which satisfy the equation Sl^=S2^, they generate a group G under which Si^ is invariant. That is, the cyclic group generated by ^i^ is composed of operators which are invariant under G, since Si^ is commutative with each Quarterly Journal of Maiketnatics, vol. 41 (1909-10), p. 169. 143 144 GROUPS DEFINED ABSTRACTLY [Ch. VI of the op>erators si and S2. From the fact that 51^ = 52^, it results that Each of the operators siS2~^, ^2^1"^ is therefore transformed into the other by each of the two operators si, S2. That is, the cyclic group generated by either of these operators is invariant under G, and each of its operators is transformed into its inverse by si as well as by S2- The abelian group generated by the two operators si^, SiS2~^ must therefore be invariant under G, and it involves either all, or just half of the operators of a When each of the two operators si, S2 is of odd order, G is cyclic, since each of these operators is equal to the other, and G is generated by ^i^ in this case. When one of these operators is of odd order while the other is of even order, G is generated by the operator of even order and 51^2 ~^ is of order 2. The only case which requires further consideration is therefore the one in which the common order of Si, S2 is an even number 2. A necessary and sufficient condition that G be abelian is that the order of SiS2~^ divide 2. If siS2~^ = l, Si=S2 and G is the cyclic group generated by Si. If the order of 5i52"^ is 2, G is either the cyclic group of order 2 or the abelian group whose invariants are 2 and 2n. It remains only to consider the cases when G is non-abelian; that is, when the order of SiS2~^ exceeds 2, and hence Si, S2 have the same even order 2n. It is evident that the common order of ^i, ^2 is not limited by the relation 51^ = 52^. That is, for an arbitrary value of n we can find two operators Si, S2 of order 2w such that they satisfy the equation Si^ = S2^. In fact, the value of n is not limited if we impose the additional condition that the order of 5i52~^ shall be an arbitrary number m, since two generators of order 2 of the dihedral group of order 2m may be multiplied by an operator of order 2 which is commutative with both of these generators such that the products satisfy both of these con- ditions. In other words, for any arbitrary number pair m, n, we can find two operators ^i, S2 of order 2n such that they 56] TWO GENERATORS HAVING A COMMON SQUARE 145 have a common square and that the 'order of SiS2~^ is m. The order of the group generated by these two operators is always a divisor of 2mn and a multiple of mn, since ^i^ and siS2~^, are of orders n and m respectively and the cyclic group generated by these operators cannot have more than two common operators. It results from the preceding paragraph that G is completely defined by the three conditions whenever either w or is odd. In fact, when si, S2 fulfil these conditions and either w or is an odd number, G is the group of order 2mn obtained by establishing an {m, n) isomorphism between the dihedral group of order 2m and the cyclic group of order 2w. If m and n are both even, G may again be con- structed by establishing such an {m, n) isomorphism when the order of G is 2mn. WTien this order is only mn, G may be constructed by establishing a simple isomorphism between n cyclic groups of degree and order w, and then extending this group by means of an operator of order 2n which transforms into its inverse each operator of this cyclic subgroup, permutes its systems of intransitivity, and has its wth power in this sub- group. That is, the two operators 5i, ^2 may he so selected as to fulM the three conditions Sl^=S2^ (5l52-^)* = l, 51^ = 1 and to generate either of two groups when m and n are both given even numbers. When at least one of them is a given odd number, the group generated by Si, 52 is completely determined by the three given relations. To illustrate this theorem we begin with the case when m = 4 and = 2. The group of order mn in this case is clearly the quaternion group, while the group of order 2mn may be constructed by establishing a (4, 2) isomorphism between the octic group and the cyclic group of order 4 so as to obtain the group of order IG involving 12 operators of order 4 which have only two distinct squares. This group has the quaternion 146 GROUPS DEFINED ABSTRACTLY [Ch. VI group for a quotient group but not for a subgroup. If a set of generating operators satisfy certain conditions, the largest group which they may generate has all the other groups which they may generate for quotient groups. This follows directly from the theory of the quotient group. When m = n = 2, G is clearly abelian and is of order 4 or 8. In the former case it is cyclic and in the latter it is of type (2, 1). When m = 3 and w = 2, G is the dicyclic group of order 12. When n = l, the category of groups under consideration clearly coincides with the dihedral groups. When n = 2 and m is odd it coincides with all the dicyclic groups whose orders are not divisible by 8, and when w = 2 and m is even, the groups of order inn coincide with the totality of the dicyclic groups^ while those of order 2mn may be obtained by establishing an (w, 2) isomorphism between either the dihedral group of order 2m or the dicyclic group of order 2m and the cyclic group of order 4. Hence the dihedral groups and the dicyclic groups may both be regarded as special cases of groups generated by two operators having a common square. Since ^1^2 = SiS2~^S2^, it results that the order of the product of the two operators ^1 and 52 is either the least common multiple between m and , or it is exactly half this least common multiple. That is, the order of the product of two operators is not restricted by the fact that they have a common square, and the order of the group which they generate is always a divisor of the double of the square of the order of this product. If we let /i=5i and t2 = siS2~^, it results that the group (/i, 12) is identical with (51, 52). That is, every group that can be generated by two operators having a common square can also be generated by two operators such that the one transforms the other into its inverse, and vice versa. Hence we have two abstract definitions for this category of groups. The latter definition is more convenient than the former for the purpose of obtain- ing directly the abstract properties of these groups, but in the abstract theory these groups frequently present themselves under the former definition, and hence it is very important to know that the two given definitions apply to the same cate- 57] GROUPS OF THE REGULAR POLYHEDRONS 147 gory of groups. It results directly from the given equation that S2 = t2'Hi. That is, if trH2ti=t2-^ then ti^ = (t2-Hiy. In the special case of the dihedral groups the two equivalent definitions reduce to EXERCISES 1. If Si, Si are two operators, neither of which is the identity, which satisfy both of the conditions: they must also satisfy the conditions: Si~^S2Si = S2^, S2~^SiS2=Si*; and they generate either the quaternion group or the four-group. 2. Find the number of operators of order 6 in the group of order g which is generated by Si, S2 subject to the following conditions: 5i= 1, Si~^SiSi=S2~^. Prove that the central of this group is either of order 3 or of order 6, and that g may be an arbitrary multiple of 6. 67. Groups of the Regular Polyhedrons.* The regular tetrahedron evidently admits two movements of order 3 whose product is of order 2. If we represent these movements by Si, S2 respectively, these operators must fulfil the conditions 5i3=52^ = (5i52p = l, or 51^=52^=1, SiS2 = S2^Si^. We proceed to prove that these conditions define a group of order 12; whence, as the group of movements of the regular tetrahedron is evidently of order 12, the group defined by the given two equivalent sets of conditions must be the group of the regular tetrahedron. To prove that (si, S2) is of order 12 we may proceed as follows: The three conjugate operators of order 2 SlS2, S2S1, Si^S2Sl^ are commutative, since 525 1%2 = Si^S2^ ' S2^S\^ = S\^S25\^. * These groups were first studied by means of abstract definitions by W. R, Hamilton, cf. Bibliotheca Mathemalka, vol. 11 (1910-11), p. 314. 148 GROUPS DEFINED ABSTRACTLY (Ch. VI As two operators of order 2 whose product is of order 2 gener- ate the four-group, it results that {si, S2) involves the four- group represented by the operators 1, S1S2, S2S1, Si^S2Si^. As this subgroup is invariant under si as well as under 51^2, it must be invariant under 52, and therefore also under (^i, 52). Hence the group generated by Si and this subgroup of order 4 is of order 12. Since this group involves ^1 and 51J2, it must also involve 52; that is, it must be {si, 52). This proves the theorem : // the order of the product of two operators of order 3 is of order 2 they generate the tetrahedral group. If we replace Ji by /i and s\S2 by t2, it results that /i, ti-'^ti are two operators of order 3 whose product is of order 2; hence {t\, tr^ti) is the tetrahedral group. Since (/i, ti-^t2) = {ti, /2) it results also that if the order of the product of two operators of orders 2 and 3 respectively is 3, then these operators must generate the tetrahedral group. That is, Si, S2 generate the tetrahedral group if they fulfil either one of the following two sets of conditions: Si^ = S2^ = (SiS2y = 1, Si^=S2^ = (SiS2y = 1. These two sets of equations furnish two very useful definitions of this important group. The group could also be defined by the facts that its order is 12 and that it does not involve a sub- group of order 6, as well as by the facts that it is of order 12 and contains four subgroups of order 3. The cube is clearly transformed into itself by 24 movements of rigid space, and the order of each of these movements is equal to one of the four numbers 1, 2, 3, and 4. It is not diffi- cult to find, among these 24 movements, two of orders 3 and 4 respectively whose product is of order 2. If we represent these tv;o movements by Si, 52, they must therefore satisfy the equa- tions 5l3=52* = (5i52)2 = l. 67) GROUPS OF THE REGULAR POLYHEDRONS 149 We proceed to prove that these conditional equations define a group of order 24, and hence they must define the group of the cube. To prove that they define a group of order 24 we may proceed as follows: The group (51, S2^) is the tetrahedral group, as results directly from the fact that {SiS2^y = 5i52 S2S1S2 S2S1S2 So = S1S2 'Si^'5i^-S2 = l. This tetrahedral group is invariant under 52, since S2~^SiS2 = 52%l52 = 52^ S2S1S2 = So^Si^. As (5 1, S2^) is invariant under S2 and involves S2^, it follows that (^1, S2) is of order 24. That is, Two operators of orders 3 and 4 respectively whose product is of order 2 generate the group of the cube. From the given equations it results directly that this theorem may also be stated in either of the following two ways: Two operators of orders 2 and 3 respectively whose product is of order 4 generate the group of the cube, or two operators of orders 2 and 4 respectively whose product is of order 3 generate the group of the cube. The group of the cube may therefore be defined as the group generated by 5 1, 52 when these operators fulfil any one of the following four sets of equations: Sl^=S2^ = {SiS2Y = \; 51^ =52'* = !, SxS2=S2V\ 5i2 =523 = {s^SiY = 1 ; Sx^ =52^ = {siSiY = 1. The group of the cube is also known as the octahedron group, in view of the fact that the group of movements of the regular octahedron coincides with that of the cube. This group presents itself in many investigations and has been defined abstractly in a number of other ways. Among these are the following: Two operators of order 4 generate the octahedron group pro- vided their product is of order 3 and each of them is trans- formed into its inverse by the square of the other. The octa- hedron group is the smallest group that can be generated by two operators whose orders exceed 2 and which are such that each is transformed into its inverse by the square of the other.* * Annds of Mathematics, vol. 21 (l'J07), p. 50. 150 GROUPS DEFINED ABSTRACTLY ICh. VI The octahedron group is completely defined by the fact that it can be generated by three cyclic non-invariant subgroups of order 4 which do not involve a common subgroup of order 2 nor generate other operators of order 4.* The octahedron group may be defined as the group which contains exactly 9 operators of order 2 such that the order of the product of any two does not exceed 4, while there are at least two such opera- tors whose product is of order 4.t 58. Group of the Regular Icosahedron. Both the regular icosahedron and the regular duodecahedron admit two move- ments of orders 2 and 5 respectively whose product is of order 3. If we let 5i, 52 represent these movements it results that these solids are transformed into themselves by operators which fulfil the following equations: It is not diflScult to prove that these equations define the simple group of order 60 4 As the regular icosahedron and the regular duodecahedron admit exactly 60 movements, it will result from this proof that their group of movements must be the simple group of order 60. Hence this group is frequently called the icosahedron group. To prove that (^i, S2) is of order 60 if no restrictions are placed on these operators except those implied in the given equations, we may proceed as follows: The operators siS2^siS2^Si, siS2^siS2^si are of order 2, since they coincide with their inverses. They transform S2 into its inverse, since the product of S2 and either one of these opera- tors is of order 2, as may be seen directly if we employ the equation 5i525i=52^5iS2*, resulting from (si52)^ = l. From this equation it results also that {SiS2^)^ = {siS2^)^ = l. In fact, {SlS2Si)^=S2^{SlS2^)^S2=l, sinCC {815281)" = SlS2''Sl. * Mathematische Annalen, vol. 64 (1907), p. 344. t American Journal of Malhcmalks, vol. 29 (1907), p. 8. t Hamilton, Philosophical Magazine, vol. 12 (1856), p. 44fi. 58] GROUP OF THE REGULAR IC08AHEDR0N 151 As SiS-2^ is the transform of the inverse of siSi^, the given state- ment is proved. The most general product that can be formed with the operators Si, S2 is of the form If the number of factors in this product exceeds six it can evi- dently be reduced by means of one or more of the following equations: SlS2Sl=S2'^SiS2'^, SiS2'^Si=S2SlS2, (^1^2^)^= 1, SiS2^Si = S2*SiS2^SiS2'^, SiS2^Si = S2SlS2^SiS2. From these equations and from the fact that SiS2^SiS2^SiS2' = S2~SiS2^SiS2^Si it results that all the distinct operators of (51, S2) can be written in one of the following forms: 52", S2'^SiS2'', S2'^SiS2^SiS2*', S2'"5i52%lS2^5i(w, = 1, 2, . . . , 5). Hence (51, S2) is of order 60 if we assume that si, S2 fulfil the equations given above. The three sets of equations Sl^=S2^={siS2y = l, Si^=S2^ = {siS2)^=l, Si^ =$2^ = {5152^ = 1 are evidently equivalent and hence each of them defines the icosahedral group. In fact, if we should assume that the opera- tors 5i, 52 merely satisfy any one of these sets of equations, they would generate the icosahedron group unless both of them were the identity. Hence we have the theorem: // the three numbers 2, 3, 5 are the orders of two operators and of their product, these operators must generate the icosahedron group irrespective of which of these three numbers is the order of the product. EXERCISES 1. If two operators merely satisfy the equations defining the tetra- hedral group they may generate the cyclic group of order 3, and if they merely satisfy the equations defining the octahedral group they may generate the symmetric group of order 6 or the group of order 2. 2. If a group of order 12 contains no invariant operator of order 2 it must be tetrahedral. 152 GROUPS DEFINED ABSTRACTLY (Ch. VI 3. If two operators of or^er 3 have a product of order 2n whose square is invariant under both of these operators, they generate a group of order 12/1 whose group of cogredient isomorphisms is the tetrahedral group. 69. Generalizations of the Group of the Regular Tetra- hedron.* An immediate generalization of the tetrahedral group is given by the equations It results directly that 5i^ = / is invariant under (5i, 52) and that the group of cogredient isomorphisms of (51, S2) is the tetra- hedral group. From the fact that S1S2 and 52^1 are two opera- tors of order 2 it results that siS2^si is transformed into its inverse by each of the operators S1S2 and 52^1 . Moreover, {SiS2^Siy = SiS2^Si^S2^Si^Si~^ = Si.S2~^Si-^S2~^Si-^ .Si-^Si^^ = t* is both invariant and also transformed into its inverse under (51, S2). Hence the order of / is a divisor of 8. If this order is 8, the order of G is 96. That there is such a group of order 9G may readily be seen by means of the two substitutions 5i =ac'ce'eg'ga''hm'nfoj'hk'ld'mp'fi'jh'kn'do'ph'U', S2 = am' je'on'gk'pc'ml'ei'na'kj'co'lg'ip' hd'dJ'Jh'W. The existence of this group of order 96 and some of its prop- erties may also be established abstractly as follows: Let ^'1, ^'2 be two operators of order 3 whose product is of order 4, and suppose that they have been so chosen that (51', 52') is the group of order 24 which does not involve any subgroup of order 12, i.e., the non-twelve f ^"24. If we extend this group by means of an operator t of order 8 which is commutative with each of its operators and generates its operators of order 2, it is clear that s\t, S2t may be used for S\, S2 respectively. That such an operator of order 8 actually exists can readily Some of these developments could have been presented more briefly by employing the theorem that if the square of an element of a commutator is com- mutative with the commutator it transforms this commutator into its inverse. The presentation adopted seems a little more elementary'. t American Journal 0} Mathematics, vol. 32 (1910), p. 65. 5Ul GENERALIZATION OP TETRAHEDRAL GROUP 153 be seen if the non-'twelve G24 is written in the regular form in four distinct sets of letters, and a simple isomorphism is estab- lished between these groups. The operator of older 4 which merely permutes cyclically the corresponding letters in these four constituent groups, multiplied by the operator of order 2 in one of these constituents, is clearly the operator of order 8 in question. When t is of order 4, G is the direct product of the tetra- hedral group and the cyclic group of order 4, since Si*, 52* are two operators of order 3 whose product is of order 2. When t is of order 2, G is the direct product of the tetrahedral group and the group of order 2, since {si^S2^)^ = Si^^ = l. Combining these results we arrive at the theorem : // two non-commutative operators s\, $2 satisfy the equations si^ = S2^, {siS2)- = \, they generate a group of order 96, or the direct product of the tetrahedral group and one of tite following three groups: tlie cyclic group of order 4, the group of order 2, tJie identity. Hence there are four and only 4 non-abelian groups which may be generated by two operators which satisfy these two conditions. The cyclic group of order 12 and its subgroups are evidently the only abelian groups which can be generated by two operators satisfying these conditions, and all of these groups can be generated by two such operators. Another generalization of the tetrahedral group is furnished by the equations Si^ = S2^, (5152)^ = 1. The cyclic group generated by s\^ = t is again invariant under G, and, as {s\-^S2S\-S2~^y = {siS2Si'^s\-^'^Y=Si-^^ is both in- variant under G and also trahsformed into its inverse by So, it results that the order of / is a divisor of 10. If we assume that / is of order 10, G is of order 120. We proceed to prove that this G is the direct product of the non-twelve group of order 24 and the group of order 5. If / is of order 10 the order of S\ may be assumed to be 30, since G would be abelian if the order of si were not divisible by 3. Hence we may assume 154 GROUPS DEFINED ABSTRACTLY (Ch. VI that Si, 52 are of orders 30 and 20 respectively while Si'', $2^ are two operators of orders 6 and 4 respectively whose product is of order 6, since As (si^)^ = {s2^)^ = isi^S2^)^ is an operator of order 2, it results that (51*, 52*) is the non-twelve group of order 24, and that G is the direct product of this group and the group of order 5. When / is of order 5, G is evidently the direct product of the tetrahedral group and the group of order 5, and, when / is of order 2, G is the non-twelve group of order 24. This completes a proof of the theorem : // two non-commutative operators si, S2 satisfy the two condi- tions 5i^ = 52^, (5152)^ = 1, they must generate one of the following four groups: the tetrahedral group, the non-twelve group of order 24, the direct products of these respective groups and the group of order 5. If 5i, 52 are commutative they must generate the group of order 15 or a subgroup of this group, and they may be so chosen as to generate any one of these subgroups. 60. Generalization of the Octahedron Group. The equations 5l2 = 52^ (5152)^ = 1 evidently furnish a direct generalization of the octahedron group. To obtain an upper limit for the order of such a group we may consider the two operators 5i, 52~^5i52. They have a common square and this square is invariant under G. Hence it results that the commutator of S\, S2 is transformed into its inverse by each of the operators S\, 52~^5i52. Squaring this commutator there results 5i - ^52 ~ ^5i52 5i - ^52 ~ ^5i52 = 5i - 85i52^5i525i52^5i52 = 5i - 5i52^ 52 " ^5i - ^52 ~ *5i - * 525i52 = 5i - ^^5i525i52%l525i52 = 5i-^'*5i525i525i-^52"^5i-^=5i -20 -52-^51 -^5251 = 51-20. (5i-l52-^5i52)-^ 601 GENERALIZATION OF THE OCTAHEDRON GROUP 155 Hence it follows that {si-^S2-^siS2)^ = si-^^. As Si~^^ is trans- formed into its inverse by ^i, the order of si is a divisor of 40 and the order of G must therefore be a divisor of 480. We begin with the case when Si is of order 8 and hence G is of order 96. To prove the existence of this G we may extend the non-twelve G24 by means of an operator of order 8 which transforms it according to an operator of order 2 correspond- ing to an outer isomorphism. Under the resulting group of order 96 the given G24 must therefore be transformed accord- ing to the octahedral group, which is the group of isomorphisms of this G24 as well as of its quaternion subgroup. If si' is the operator of order 2 in this octahedral group, which corresponds to the given operator Si of order 8 in the group of order 96, and if ^2' is an operator of order 3 in this octahedral group such that Si'so' is of order 4, we can select an operator 52 of order 12 corresponding to $2' so that 51^ = 52^, (5iS2)* = l; for, all the operators of this Gge which correspond to operators of order 4 in the octahedral group are of order 4, since the commutator subgroup of each of the three Sylow subgroups of order 32 is a subgroup of order 4 contained in the invariant quaternion subgroup. It is easy to verify that this group of order 96 is also generated by the following substitutions: Si = aa'gg'ee'cc' bo'hm'fk'di' id'oh'tnh'kf -jn'pl'nj'lp' , S2 = amlgkjeipcon bdfh a'm'l'g'k'j'e'i'p'c'o'n' h'd'J'h'. Having established the existence of a group of order 96 generated by two operators of orders 8 and 12 which satisfy the given conditions, it is easy to find the group of order 480 generated by two operators of orders 40 and 60 respectively which satisfy these conditions. In fact, it is obvious that such operators exist in the direct product of the given G96 and the cyclic group of order 5. For if 5i, 52 generate the former group and satisfy the relations si^ = S2^, {s\S2Y = \, and if / is an operator of order 5 which is commutative with each of these operators, then s\t, S2t~^ will also satisfy these equations. Hence we have proved that the largest group which can be generated by two operators which satisfy the two conditions 156 GROUPS DEFINED ABiiTRACTLY (Ch. VI Si^=S2^, (si52y = l, is tjie direct product of the group of order 5 and the group or order 9G obtained by extending the non- twelve group of order 21 by means of an operator of order 8 which transforms it according to an outer isomorphism of order 2. It is evident that the group of order 48 obtained by estab- lishing a (2, 12) isomorphism between the cyclic group of order 4 and the octahedron group is generated by two operators of orders 4 and 6 respectively which satisfy the given equations. Moreover, the direct product of this group of order 48 and the group of order 5, and the direct product of the octahedral group and this group of order 5 contain two generating opera- tors which satisfy the conditions under consideration. Hence we have arrived at the theorem: There are exactly six non-ahelian groups which can he generated by two operators which fulfil the equations 51^ = 52^, (5152)'* = 1. Three of these are of orders 24, 48, and 96, respectively, while the other three are the direct products of these respective groups and the group of order 5. A second generalization of the octahedron group is given by the equations SX^=S2\ (5152)2 = 1. Since the two operators 5i52, 525 1 are of order 2, they generate a dihedral group. To determine an upper limit of the order of this group we observe that (5251^52)^ = (52^5152^)^ -51^=5121. As sr^ is invariant under G, its order is a divisor of 42, and an upper limit of the order of this dihedral group is evidently 12, while the order of G is a divisor of 336. When s\ is of order 6 the order of G is 48. Moreover, it is easy to see that the group of order 48, which may be obtained by extending the non- twelve group of order 24 by an operator of order 2 which trans- forms it according to an outer isomorphism, is generated by two operators of orders 6 and 8 respectively, which satisfy the given conditions. This group may be represented transitively on eight letters, and it involves 12 operators of each of the orders GO] GENERALIZATION OF THE OCTAHEDRON GROUP 157 2 and 8 in addition to the given subgroup of order 24. It is the group of isomorphisms of the non-cyclic group of order 9. If we multiply the two given generators by an operator of order 7 and its inverse, the operator of order 7 being commuta- tive with each of these generators, we obtain two operators of orders 42 and 56 respectively which satisfy the conditions in question, and hence we have the theorem: // two operators satisfy the conditions Si^ = S2*, (5152)^ = 1, the largest group which they can generate is the direct product of the group of order 7 and the group of isomorphisms of the non-cyclic group of order 9. The total number of the non-ahelian groups which can he generated by two operators which satisfy these equa- tions is six: viz., the dihedral group of order 6, the octahedral group, the group of isomorphisms of the non-cyclic group of order 9, and the direct products of these respective groups and the group of order 7. The third generalization of the octahedron group to be considered is given by the equation We may again consider the commutator of 5i, ^2 and observe that {Sl - ^S2 ~ ^SiS2)^ = (525152^)^51^ = 5i%2 -'^{s2^SiYs2^ = 5i652-2(5i452-^5i-0W = 5i*. As 5i^ is transformed into its inverse by 5i, it results that the order of 5i is a divisor of 36, and hence the order of G is a divisor of 432. It is easy to see that the group of order 48, which may be constructed by extending the non-twelve group of order 24 by means of an operator of order 4 which has its square in this non-twelve group and transforms it according to an outer isomorphism of order 2, can be generated by two operators of orders 4 and 8 respectively which satisfy the given conditions. If we multiply this 5i and this 52 by an operator of order 9 and by its fifth power respectively, this operator of order 9 being commutative with each of the operators 5i, 52, and having only the identity in common with (51, 52), we obtain two operators 158 GROUPS DEFINED ABSTRACTLY (Ch. VI of orders 36 and 72 respectively which satisfy the given con- dition and generate the group of order 432 in question. Hence it is easy to deduce the following theorem : I J two non-commutative operators satisfy the conditions Si^=S2*t (^1^2)^ = 1, they may generate the dihedral group of order 6, the octahedral group, the group of order 48 obtained by extending the non-twelve group of order 24 by means of an operator of order 4 which has its square in this group and transforms it according to an outer isomorphism of order 2, the direct products of these respec- tive groups and a cyclic group of order 3 or 9. Hence there are exactly nine non-abelian groups which may he generated by two such operators. EXERCISES 1. There are exactly six non-abelian groups whose two generators Si, St satisfy the equations Si^=Si^, (5152)^=1. They are the icosahedron group, a group of order 120, and the direct products of these respective groups and the cyclic groups of orders 5 and 25. 2. If two commutative operators satisfy the equations 5i*=52*, (51^2)'= 1, they generate a cyclic group whose order is 3, 7, or 21; if they satisfy the equations Si^=S2^, (5152)^= 1, they generate a cyclic group whose order is 2, 4, 8, or 16; if they satisfy the equations Si^=Si^, (Ji5j)*=l, they generate a cyclic group whose order is either 5 or 25. 3. There are exactly six non-abelian groups whose generators satisfy the equations Si^=Si^, {sis^^= 1. They are a group of order 1920 and the direct products of the icosahedral group and the cyclic group of order 2", a=0, 1, 2, 3, 4. 4. If two commuiative operators satisfy the equation 5i*=^2S(5352)' = 1, they generate a cyclic group whose order is :; 3, 6, 9, or 18; if they satisfy the equations Si^=S2^, {siS^*=\, they geneiate a cyclic group whose order is 2, 4, 5, 10, or 20; if they satisfy the equations Si*=Si*, {siSiy=\, they generate a cyclic group whose order is 2, 7, or 14. CHAPTER VII ISOMORPHISMS 61. Relative and Intrinsic Properties of the Operators of a Group. The operators or elements of a group have both relative and intrinsic properties. The latter relate to period- icity and are common to group operators and the roots of unity. Hence some of the earliest workers in abstract group theory associated this theory with the roots of unity. For instance, Cayley remarks that " a group may be considered as repre- senting a system of roots of the symbohc binomial equation 5" = 1,"* and Sir W. R. Hamilton regards the icosahedral group as " a system of non-commutative roots of unity which are entirely distinct from the i, j, k of the quaternion though having some analogy thereto." f He calls this group the Icosian Calculus. In a non-abelian group the relative properties of the opera- tors are of greatest interest, while they are of comparatively little interest as regards abelian groups. In fact, they reduce to the question of common subgroups generated by these opera- tors in the latter case. In an automorphism of a group, the corresponding operators must evidently have the same intrinsic as well as the same relative properties. The great importance of the study of automorphisms rests on the fact that the proper- ties of an operator are the same as those of the operators which correspond to it in some autoniorphism of the group, and hence these properties need to be studied for only one of these corre- sponding operators. Thus the concept of isomorphisms econo- mizes thought, which is a fundamental object in mathematics. * Cayley, Philosophical Magazine, vol. 7 (1854), p. 40. t Hamilton, Philosophical Magazine, vol. 12 (1856) p. 446. ' 159 160 ISOMORPHISMS ICh. VII In its most elementary form the concept of isomorphisms is one of the oldest in mathematics, as it lies at the base of the development of abstract numbers. The concept of abstract number evidently rests on a kind of isomorphism between con- crete units of various kinds, so that for many purposes we may fix our attention entirely on what is common, viz., the abstract concept of units. In the theory of groups the con- cept of isomorphisms assumes a new importance in view of the fact that the different automorphisms of a group may be repre- sented by the corresponding substitutions on its operators, and, as was noted in 19, the totality of these substitutions con- stitute a group known as the group of isomorphisms,* or the group of automorphisms, of the original group. We have thus associated with each group its group of isomorphisms, which is of fundamental importance in many applications of the group. 62. Group of Isomorphisms as a Substitution Group. If g distinct letters are placed in a (1, 1) correspondence with the operators of the group G.of order g, the symmetric substitution group will correspond to the totality of the possible arrange- ments of the operators of G. Such an arrangement cannot correspond to an automorphism of G unless the identity corre- sponds to itself. Hence the group of isomorphisms I oi G can always be represented as a substitution group on at most g 1 letters, and its order must therefore be a divisor of (g 1)!. This order cannot be equal to {gl)\ except in caselpf three groups besides the identity, viz., the groups of orders ''2 and 3, and the four-group. In fact, it is evident that / cannot be more than doubly transitive on g 1 letters, since the correspondence of two operators fixes the correspondence of their product. In particular, the order of the group of isomorphisms of any finite group is a finite number. A necessary condition that / be transitive on g 1 letters is that all the operators of G besides the identity have the same Isomorphisms were first studied in an explicit manner by C. Jordan and A. Capelli. Their group properties were first studied by O. Holder and by E. H. Moore. 62] ISOMORPHISMS REPRESENTED BY SUBSTITUTIONS 161 order, and hence g must be of the form />", p being a prime number. Since the correspondence of two operators determines the correspondence of their powers, it is clear that / cannot be a primitive substitution group unless p = 2. If all the opera- tors of G besides the identity are of order 2, / is evidently doubly transitive. Hence it results that a necessary and suf- ficient condition that / be primitive on g l letters is that all the operators of G be of order 2.* The group / is generally intransitive on gl letters, and the number of its systems of intransitivity is equal to the number of complete sets of conjugate operators of G under /. In par- ticular, the number of characteristic operators of G is equal to g diminished by the degree of /. A sufficient condition that / is simply isomorphic with one of its transitive constituents, when it is represented as such a substitution group, is that G is generated by one of its complete sets of conjugates under /. When G is abelian it is generated by its operators of highest order, and these constitute a complete set of conjugates under /. As they constitute the only complete set of such conjugates that generate G, it results that the group of isomorphisms of an abelian group can always be represented in one and in only one way as a transitive substitution group on letters corresponding to operators of this abelian group. In other words, if the group of isomorphisms of an abelian group is represented on letters corresponding to the various operators of this abelian group, this group of isomorphisms has only one transitive constituent which is simply isomorphic with it, since every abelian group of order />*", except the group of order 2, admits a non-identical isomorphism in which every operator which is not of highest order corresponds to itself. A like theorem does not apply in general to the non-abelian. groups. In fact, if the / of a non- abelian group is represented on the ^1 letters corresponding to the operators of the group, excepting the identity, the number of its transitive constituents which are simply isomorphic with / may vary from zero to an indefinitely large number, as results from the alternating groups. * E, H. Moore, Bulletin of the American Mathematical Society, vol. 2 (189f), p. 33. 162 ISOMORPHISMS (Ch. VII Suppose that G is abelian and that / is represented as a transitive group on letters corresponding to operators of G. These operators must be composed of the operators of highest order (m) in G. As any operator of highest order may be re- garded as an independent generator of G, it is evident that / cannot be regular unless G is cyclic and that / is always regu- lar when G is a cyclic group. As an abelian group cannot be represented as a non-regular transitive substitution group, it results directly from this fact that A necessary and sufficient condition that the group of iso- morphisms of an abelian group be abelian is that this abelian group be cyclic* The subgroup composed of all the substitutions of I which omit one letter must omit exactly <t){m) letters, <}>(m) being the totient of m, since an operator of order m generates <f>{m) oper- ators of this order. As the number of substitutions which are conlmutative with every substitution of a transitive group of degree n is a, where a is the exact number of the letters omitted by the subgroup of this transitive group which is composed of" all its substitutions which omit a particular letter, it results directly that / contains exactly <^(w) invariant substitutions, and that these substitutions transform every substitution of / into the same power. Hence the theorem: // an abelian group involves operators of order m but none of higher order, its group of isomorphisms contains exactly ^{m) invariant operators. 63. Groups of Isomorphisms of Non-abelian Groups. Sup- pose that G is non-abelian and that / is represented on letters corresponding to a set of operators of G. If all of these opera- tors were commutative they would generate a characteristic abelian subgroup of G. If this subgroup were in the central of G, I would involve operators corresponding to inner auto- morphisms, but which would not afifect any of the operators of the given set. This would also be the case if this subgroup * This theorem and the theorem of the following paragraph were proved abstractly in 41. The fundamental importance of these theorems seems to justify the present alternative proofs. 5 63l ISOMORPHISMS OF NON-ABELIAN GROUPS 163 were not in the central of G, since one of these operators would then transform into difTerent operators some of the operators of G which are not contained in this set. Hence it has been proved that The group of isomorphisms of a non-ahelian group cannot he represented on letters corresponding to a set of relatively com- mutative operators of this non-ahelian group. It is now easy to see that if the group of isomorphisms 7 of any group G can be represented transitively on letters corre- sponding to a set of operators of G, then / cannot be more than doubly transitive. In fact, when G is non-abeUan and we fix the correspondence of two non-commutative operators of the set in an automorphism, the correspondence of a third operator has also been fixed, since such an operator may be obtained by transforming one of the two given operators by the other. That is, When the group of isomorphisms of any group G is represented on letters corresponding to operators of G, then this group of iso- 'morphisms is at most douhly transitive. It is very easy to find non-abelian groups in which the / is doubly transitive if it is constructed in a given manner. As such a group we may consider the metacyclic group of degree p and of order p{p \). The / of this group can evidently be represented on the p letters corresponding to its operator of order 2, and if it is represented in this way it coincides with the metacyclic group itself. Instead of representing I on the operators of a group we may frequently represent it more con- veniently on a set of generating subgroups. If this is done I may be more than doubly transitive, as results from the fact that the group of isomorphisms of the tetrahedral group is the symmetric group of degree 4 if each letter corresponds to a sub- group of order 3. If / is a primitive group on letters corresponding to a set of operators of a non-abelian group G, its invariant subgroup corresponding to the inner isomorphisms of G must be transi- tive, since a primitive group cannot involve an intransitive in- variant subgroup. That is, if 7 is a primitive group on letters 164 ISOMORPHISMS [Ch. VII corresponding to the operators of G, then G is either the abelian group of order 2" and of type (1,1, 1, . . . ) or G transforms transitively the set of operators to which the letters of / corre- spond. As every multiply transitive group is primitive, this theorem applies to all multiply transitive groups as well as to the simply transitive primitive groups. EXERCISES 1. If G is the symmetric group of degree 4 its / may be represented as an imprimitive group of degree 6 on letters corresponding either to its operators of order 4, or to its six conjugate operators of order 2. These two imprimitive groups are not conjugate as substitution groups, since the one is composed of positive substitutions, while the other contains negative substitutions. 2. The group of isomorphisms of the symmetric group of degree n, n^6, can be represented as a transitive substitution group on the n(n 1)/2 letters corresponding to the transpositions of the symmetric group. When / is thus represented, it is a simply transitive primitive group whenever n>4. Suggestions: Use the theorem that the symmetric group of degree n, ?^6 and n>2, is its own group of isomorphisms, and that it has no outer isomorphisms. See 65. 3. Prove that if the / of the quaternion group is represented as a sub- stitution group whose letters correspond to its operators of order 4, it will be conjugate with the group in the first of these exercises which involves negative substitutions. 64. Doubly Transitive Substitution Groups of Isomorphisms. If / is doubly transitive on letters corresponding to operators of G, each of these operators generates a cyclic subgroup (s) which is transformed into itself under the holomorph of G by a subgroup composed entirely of operators which are com- mutative with s; for, if a complete set of conjugate operators of G under / includes at least two powers of the same operator, the operators of this system must be transformed according to an imprimitive group. Suppose that si and S2 are two oper- ators of G which correspond to letters of I. We may assume that 5i, 52 are non-commutative; for, if all such operators were commutative, G would be abelian and hence the order of every operator of G would divide 2. Since this case is so elementary, 64] DOUBLY TRANSITIVE GROUPS OF ISOMORPHISMS 165 we shall exclude it in what follows and hence we shall assume that si, 52 are non-commutative. If si, 52 correspond to themselves in a given automorphism of G, all the operators of the subgroup generated by 5 1, 52 must also correspond to themselves and this subgroup must include more than two operators which are conjugate to 5i, 52 under /. Hence we have as a first result: // the group of isomorphisms of a group G can he represented as a doubly transitive group on letters corresponding to operators of G, then the subgroup composed of all substitutions which omit one letter of this doubly transitive group is eitlier imprimitive or it is a regular group of prime degree. This theorem follows directly from the well-known theorem that the subgroup which is composed of all the substitutions which omit one letter of a non-regular primitive group of degree n is always of degree w 1. When G is abeUan the given theorem evidently remains true and the imprimitive group in question involves systems of two letters each except when G is the four- group. When the subgroup which is composed of all the substitu- tions of / which omit one letter is a regular primitive group, the order of / is pip-^-l), p being a prime, and / involves p-{-\ subgroups of order p. It must therefore involve an invariant subgroup of order />+l which involves p conjugate operators under /. That is, the subgroup of order />+l must be the abelian group of order 2^ and of type (1,1,1,...). Hence the following theorem: // / is doubly transitive on letters corresponding to operators of G and if the subgroup composed of all the substitutions which omit one letter of I is primitive, then I is of order />(/>+!), P being a prime, and it involves an invariant subgroup of order p-\-l- When / is a doubly transitive group on letters corresponding to a set of conjugate operators of G, either all the operators of this set are commutative or no two of them are commutative. This results immediately from the fact that when / is doubly transitive any two of its letters can be transformed into an 166 ISOMORPHISMS [Ch. VII arbitrary pair, but a pair corresponding to two commutative operators could not be transformed into a pair corresponding to two non-commutative ones. Hence it results that when / is a doubly transitive group on letters corresponding to a set of operators of G, then G is either an abelian group of order 2* and of type (1, 1, 1, . . . ) or no two of the operators of the set under consideration are commutative. 65. Groups of Isomorphisms of the Alternating and the Symmetric Groups. In this section we propose to prove the theorems that the alternating and the symmetric group of degree n, W7^3, have the same group of isomorphisms, and that this group coincides with the symmetric group whenever w>3, with the exception of the single case when n = Q. In this special case the group of isomorphisms of the symmetric and alternating group is a well-known group whose order is 1440; that is, this order is twice that of the corresponding symmetric group. The proof of these theorems entails the proof of several auxiliary theorems, which are also of considerable interest in themselves and of still greater historic interest in view of the fact that they relate to one of the oldest problems of group theory, viz., the determination of subgroups of small index under the symmetric and alternating groups. This is known as Bertrand's problem. As it will be desirable to use the theorems that the alternat- ing group of degree n, n9^Q, involves only one subgroup of mdex n, viz., the alternating group of degree n1, and that the symmetric group of degree n involves no subgroup of index , ?^6, besides the symmetric group of degree n1, we shall establish the somewhat more general theorem, sometimes called Bertrand's theorem,* that the symmetric group of degree n, n>4, has no subgroup whose index lies between n and 2, and that its only subgroups of index n, np^Q, are of degree w 1; moreover, the alternating group contains no subgroup of index Serret, Algibre sup^rieure, 1849, p. 267. Bertrand proved this theorem in 1845, Journal de VEcole Poly technique, p. 129, by assuming the theorem, after- wards proved by Cebysev, that there is always at least one prime number be- tween /2 (exclusive) and n 2 (inclusive) whenever the natural number n exceeds 6. 65] ISOMORPHISMS OF ALTERNATING GROUPS 167 less than , >4, and its only subgroup of this index is the alternating group of degree n 1, when nj^Q. We begin with the proof of the latter part of this theorem, since the former part can be readily deduced from the latter. As the theorem of Cebysev, to which we have just referred, applies only to all numbers greater than 6, and the groups of degree seven are well known, we shall assume that n>7, and prove that the alternating group of degree n does not contain any subgroup whose index is less than +l, with the exception of its alternating subgroups of degree 1 and of index n. If such a subgroup existed it would be transitive on its letters, since the order of an intransitive subgroup could clearly not exceed 2'(n 2)!. As the order of an imprimitive subgroup is evidently less than this number, the subgroup in question would be primitive, and hence its order could not be divisible by the highest power of 3 which divides n!, since a primitive group of degree n does not involve a substitution of the form abc unless it is the alternating group of degree n. Since the order of the subgroup in question would not be divisible by the highest power of 3 that divides n\, this order would have to be divisible by the prime p, where n/2<p^n 2. Hence this subgroup would involve l-{-kp conjugate cyclic subgroups of order p. If two such subgroups had less than pl common letters, we could transform one by an operator of the other so as to obtain two such subgroups having a larger number of common letters without having all letters in common. This process could be repeated until two subgroups of order p would be found having p l common letters, and hence the primitive subgroup in question would itself involve primitive subgroups of each of the degrees p, p-\-l, ...,. It would therefore be at least four-fold transitive. As the transitive subgroup composed of all the substitutions involving a certain set of p letters would be invariant under a group of degree />+3, which would involve two transitive con- stituents of degrees p and 3 respectively, and as this transitive constituent of degree 3 would be the symmetric group of this degree, it results that each of the cyclic subgroups of order p 168 ISOMORPHISMS (Ch.VII would be invariant under a group having the symmetric group of degree 3 as a trafisitive constituent. As the group of iso- morphisms of the group of order p is cyclic, this would imply that the subgroup, composed of all the substitutions in the primitive group of degree n and index less than w + l, which transform a subgroup of order p into itself, would involve substitutions of the form ah or dbc. As this is impossible, it has been proved that the alternating group of degree n, n>7, cannot involve a transitive subgroup of degree n and of index less than w+l. Hence the symmetric group of degree n,n>7, cannot contain a subgroup of index less than n-\-l and greater than 2 except the symmetric group of degree n1. From what precedes, it results that the only subgroups of index n in the symmetric and the alternating groups are those of degree n 1, whenever n>7. This theorem is known to be true also as regards the groups of degree 7. From this fact and from the theorem in 67, it follows directly that the group of isomorphisms of the alternating and of the symmetric group of degree n, w> 6, is the symmetric group of this degree. EXERCISES 1. Prove that in an automorphism of the symmetric group of degree 6 substitutions of the form abc may correspond to those of the form abc-def, and that all the operators of order 3 in this symmetric group are conju- gate under its holomorph. 2. Prove that the symmetric groups of degrees 4 and 5 are complete groups, and that the alternating groups of these degrees have the corre- sjxjnding symmetric groups for their groups of isomorphisms. 3. Give an instance of a group which involves an invariant subgroup whose group of isomorphisms is larger than that of the entire group. 66. Several Useful Theorems Relating to the Groups of Iso- morphisms.* Every abelian group can be extended so that we obtain a group of twice the order of the original group, by means of operators of order 2 which transform each operator of this abeUan group into its inverse. These groups may be regarded as a direct generalization of the dihedral groups, and may therefore be called generalized dihedral groups as regards Cf. PhUosophical Magazine, vol. 231 (1908), p. 223. 5 661 SEVERAL USEFUL THEOREMS 169 the given abelian groups. If the given abelian subgroup involves operators whose order exceeds 2, the corresponding general dihedral group is non-abelian and vice versa. Let G be any non-abelian generalized dihedral group of order g and let H be the abelian subgroup of order g/2 which was extended to obtain G. In any automorphism of G the g/2 non-invariant operators of order 2 must correspond to themselves, and hence the I oi G can be represented as a substitution group of degree h, h being the order of H. It is evident that the non-invariant operators of order 2 in G can be arranged in h different ways after the automorph- ism of H has been fixed. Hence the order of the / of G is the same as the order of the holomorph of H. We proceed to prove that the / of G is simply isomorphic with the holo- morph oi H. In fact, this / can be represented as a transitive substitution group of degree // which involves an invariant regular subgroup which is simply isomorphic with H, since G can be made simply isomorphic with itself in such a way that the operators of H correspond to themselves while the remain- ing operators of G correspond to themselves multiplied by an arbitrary operator of H. These isomorphisms therefore corre- spond to a regular subgroup of order h in /, / being represented on letters corresponding to the non-invariant operators of order 2 in G, and this regular subgroup is simply isomorphic with H by construction. Moreover this regular subgroup is invariant under /, since H must correspond to itself in every automorphism of G, and this regular subgroup includes- all the substitutions of / corre- sponding to the automorphisms of G in which all the opera- tors of H correspond to themselves. From this fact it results that / must be a subgroup of the holomorph of H, and as the order of / is equal to the order of the holomorph of H it results that / is this holomorph. These results may be stated as follows: The group of isomorphisms of the generalized dihedral group of an abelian group H, involving operators whose orders exceed 2, is the holomorph of H. 170 ISOMORPHISMS [Ch. VII As a special case of this theorem we may observe that the group of isomorphisms of the dihedral group of order 2h, h> 2, is the holomorph of the cyclic group of order h. If E is any abelian group of even order, it may be extended by means of h operators of order 4 such that they have a common square, and each operator of H is transformed into its inverse by each of these operators of order 4. The group G of order 2// which can be constructed in this way will be called the generalized dicyclic group as regards H, since it reduces to the dicyclic group whenever H is cyclic. With the single exception when H is of order 2^ and of type (2, 1, 1, 1, ... ), the I of G can always be represented as a transitive substitution group on the given h operators of order 4. By exactly the same reasoning as was employed in the preceding case we see that, when E does not satisfy the given special condition, the I of this G is also the holomorph of G whenever the common square of the given h operators of order 4 is a characteristic opera- tor of E. This proves the following theorem: The group of isomorphisms of the generalized dicyclic group as regards an abelian group E, which is not both of order 2" and type {2, 1, 1, 1, . . . ), is the holomorph of this abelian group whenever the common square of the h operators of order 4 which were added to E is a characteristic operator of E. It should be observed that i7 is a characteristic subgroup of G also when it is both of order 2" and of type (2, 1, 1, 1, . . . ) provided the squares of the remaining operators of order 4 in G are not the same as those of the operators of order 4 in E. In this case, as well as in the more general case considered above, the group of isomorphisms of G is the subgroup of the holomorph of E composed of all the operators of this holomorph which are commutative with the square of the given h operators of order 4. The method of proof employed above may serve to establish a very elementary but useful theorem, which may be stated as follows: // a group G containing a characteristic subgroup E is such that automorphisms of G may be obtained by multiplying suc- cessively an operator s of G which is not in E by all the operators 661 SEVERAL USEFUL THEOREMS 171 of H while the operators of H are left unchanged, then the group of isomorphisms of G involves an invariant subgroup which is simply isomorphic with 11, whenever this group of isomorphisms can be represented as a transitive group of degree h, corresponding to the conjugates of s. Suppose that G is a complete group which involves only one subgroup of index 2, and consider the direct product of G and the group of order 2. If a group contains only one sub- group of index 2, this subgroup is generated by the square of the operators of the group, and, conversely, if a subgroup of index 2 is generated by the squares of the operators of a group, it is the only subgroup of index 2 in the group. That is, a necessary and sufficient condition that a group contain one and only one subgroup of index 2 is that the squares of its operators generate such a subgroup. Hence the squares of the operators in the direct product of G and the group of order 2 generate a characteristic subgroup of index 4 under this direct product. The / of this product involves an invariant operator of order 2 corresponding to the automorphisms in which two of the three co-sets as to the given characteristic subgroup are multiplied by the invariant operator of order 2. As the order of this / is the double of the order of G and as / involves an invariant operator of order 2 which is not in the / of G there results the theorem: If G is a complete group and contains only one subgroup of index 2, then the group of isomorphisms of the direct product of G and the group of order 2 is simply isomorphic with this direct product. As special cases of this theorem we may observe that the / of the direct product of the symmetric group of degree n, n> 2 and 5^6, and the group of order 2, is simply isomorphic with this direct product; and that the / of the direct product of the metacyclic group of order p(p l), p being an odd prime, and the group of order 2, is simply isomorphic with this direct product. If a group G is the direct product of characteristic subgroups, the / of G is evidently the direct product of the /'s of these subgroups. As an abehan group is the direct prod- 172 ISOMORPHISMS [Ch. VII uct of its Sylow subgroups, the I of an abelian group is always the direct product of the Vs of its Sylow subgroups. 67. Group of Isomorphisms of a Transitive Substitution Group. Suppose that G is a transitive substitution group of degree n which involves no subgroups of index n and degree , but involves subgroups of degree w 1. Its subgroups of degree 1 must therefore correspond among themselves in every automorphism of G, and these subgroups may be so lettered that they are transformed by every substitution in G in exactly the same manner as the letters of this substitution are transformed. From this it results that if each of the n subgroups corresponds to itself in any automorphism of G, each of the substitutions of G must also correspond to itself in this automorphism. That is, the I oi G may be represented on letters corresponding to these subgroups. As G involves subgroups of degree w 1 , it is simply isomorphic with its group of inner isomorphisms. Hence the I oi G may be represented as a transitive substitution group of degree n which contains G invariantly. This proves the following theorem: // G is a transitive substitution group of degree n which involves subgroups of degree n 1 but no subgroups of both degree n and index n, then the group of isomorphisms of G can be represented as a transitive substitution group of degree n which contains G as an invariant subgroup. As the symmetric group of degree n involves no subgroup of degree and index n when w?^6, and as it contains a subgroup of degree n l whenever w?^2, it results from the given theorem that the I of every symmetric group of degree w, except when n is either 2 or 6, can be represented as a substitution group on n letters, which contains this symmetric group. This sub- stitution group must therefore be the corresponding sym- metric group, as was proved above. In a similar way we may observe by means of this theorem that the metacycUc group of degree p a.nd of order p{p \) is its own group of isomorph- isms. These illustrations may suffice to point out the use- fulness of this theorem in the study of the groups of isomorphisms of substitution groups. S 67] ISOMORPHISMS OF TRANSITIVE GROUPS 173 When G is an intransitive group of degree n such that every subgroup which omits one letter is of degree exactly n 1, it is still true that these 1 subgroups of G are transformed by every substitution of G in exactly the same manner as the letters of this substitution are permuted. If G is such that these n subgroups of degree w 1 must correspond to them- selves in every possible automorphism of G, the group of iso- morphisms of G can again be represented as a substitution group on n letters. It is, however, not necessary that this substitu- tion group should be transitive, as may be seen by letting G represent the intransitive group of degree 7 and of order 24 obtained by establishing a (1, 4) isomorphism between the sym- metric groups of degrees 3 and 4. EXERCISES 1. There is no group whose group of isomorphisms is a cyclic group of odd order greater than 1.* 2. The quaternion group and the cyclic group of order 8 are the only two groups of this order that cannot be the groups of isomorphisms of any group, t 3. There are two and only two groups which have the symmetric groups of order 6 for their group of isomorphisms. J 4. Find the groups of isomorphisms of all the substitution groups whose degrees do not exceed 5. 5. The order of the group of isomorphisms of any abehan group is divisible by the number of its operators of highest order. A necessary and sufl5cient condition that the order be equal to this number is that the group be cyclic. 6. The number of distinct groups which have a given group of inner isomorphisms is either zero or infinity. 7. If a non-abelian group can be represented transitively only as a regular group, it cannot be the group of isomorphisms of an abelian group. * Annals of Mathematics, second series, vol. 2 (1900), p. 79. t Bulletin of the American Mathematical Society, vol. 6 (1900), p. 339. X Transactions of the American Mathematical Society , vd. 1 (1900), p. 399. CHAPTER Vm SOLVABLE GROUPS* 68. Introduction. A group is said to be solvable if, and only if, it contains a series of invariant subgroups such that the last of the series is the identity and the index of each of these subgroups under the next larger subgroup is a prime number. For instance, the symmetric group of order 24 contains an invariant subgroup of index 2, this subgroup contains an invari- ant subgroup of index 3, this subgroup is the four-group and contains an invariant subgroup of index 2, and the identity is of index 2 under this last subgroup. Hence the symmetric group of order 24 is solvable. The numbers 2, 3, 2, 2 are said to be its factors of composition. In general, the factors of composition of a group are the indices of the successive largest invariant subgroups. For example, the symmetric group of order 120 contains an invariant subgroup of index 2, but this subgroup involves no invariant subgroup besides the identity. Hence the symmetric group of order 120 is insolvahle and has 2 and 60 for its factors of composition. Every abelian group is evidently solvable. The terms solvable and insolvable as applied to groups of finite order are transferred from the theory of equations. An algebraic equation is solvable by rational processes in addition to root extractions whenever the group of the equation is solv- able and only then (cf. Part III). It should be observed that an invariant subgroup of an invariant subgroup is not neces- sarily an invariant subgroup of the entire group. For instance, the invariant subgroup of order 2 used in connection with the * In H. Weber's Lehrbuch dcr Algebra, solvable groups are called melacydic. In the present work we use the term metacyclic with its older meaning to repre- sent the holomorph of the group of order p. See p. 12. 174 68l INTRODUCTION TO SOLVABLE GROUPS 175 symmetric group of order 24 in the preceding paragraph is not invariant under this symmetric group, but it is invariant under the four-group. It may appear possible that one might obtain only prime factors of composition by one method of selecting the succes- sive invariant subgroups while another method would lead to composite factors. If this were possible the determination of solvability or insolvability of a group would sometimes require an examination of different sets of subgroups such that each is a largest invariant subgroup of the one which precedes it, and the last is the identity. That the factors of composi- tion of a group are entirely independent of the order in which the invariant subgroups in question are selected can easily be established by means of the theorem that two invariant subgroups which have only the identity in common must have the property that each operator of the one is commuta- tive with every operator of the other (cf. 29). We proceed to prove the invariance of the factors of composition of any group by means of this theorem. Let Go be any solvable group, and let the following series of subgroups, with the exception of Go, have the property that each is invariant and of prime index under the one which immediately precedes it: Go, Gi, G2, , Gp-i, Gp=l. In selecting another such series, suppose that the first a groups coincide with the first a groups in given series, but that the (a-f l)th is different. We thus have the series: Go, Gi, . . . , Ga-l, G a, . . . As both Ga and G'a are invariant under Ga-i, their cross-cut is also invariant under Ga-i- The quotient group of Ga-i with respect to this cross-cut must therefore involve two maximal invariant subgroups, corresponding to Ga and G'^, which have only the identity in common. Hence this quotient group is the direct product of these invariant subgroups of prime orders. This proves that the given cross-cut could be 176 SOLVABLE GROUPS (Ch. VIII selected for Ga+i regardless of whether Ga or G'a was selected, and hence it establishes the invariance of the factors oj com- position of any solvable group, since this invariance may be assumed for the groups G^ and G', as they are of lower order than Go is. It may be observed that the above proof is not dependent on the fact that Go was assumed to be solvable. The quotient group with respect to a given cross-cut is always the direct product of two simple groups, since the invariant subgroups are maximal, and hence the factors of composition are invari- ants as regards any group, even if this group be insolvable. Moreover, as the two factor groups of a direct product of two simple groups are evidently invariants of this direct product,* it results immediately that the following series of quotient groups Go/Gi, G1/G2, . , Gp-i/Gp = Gp-\ is an invariant of Go irrespective of whether Go is solvable or not. That is, the totality of these simple quotient groups is independent of the choice of the maximal invariant subgroups. The important theorem as regards the invariance of the factors of composition of any group was first proved by C. Jordan in Journal de M athematiques , volume 14 (1869), page 139. The fact that the given series of quotient groups is also an invariant of Go was observed by O. Holder in the M athematische An- nalen, volume 34 (1889), page 37. Instead of dej&ning a solvable group as one having only prime factors of composition, we may also dejQne it as a group which has the property that we arrive at the identity by form- ing the successive commutator subgroups. That is, if Ga is the commutator subgroup of Ga-i, a = l, 2, . . . , and if we form the series of groups Go, Gi, G2, . . . , Gx, * This is a special case of the theorem that factor groups of any direct product are always completely determined by this direct product. Cf. Remak, Crelle's Journal, vol. 139 (1911), p. 293. 69] SERIES OF COMPOSITION 177 a necessary and sufficient condition that Go be solvable is that for a finite value of X, Gx = l. It is evident that this implies that Gx-i is abelian and that the order of Ga is less than that of Ga-i whenever a^X. The given condition for the solva- bility of Go is therefore equivalent to saying that a necessary and suflUcient condition that a group be solvable is that none of its successive commutator subgroups besides the identity is a perfect group (cf. 29). That this second definition of a solvable group is equivalent to the first, follows immediately from the fact that if a group has an invariant subgroup of prime index, this subgroup must include the commutator subgroup of the group, and if the order of the commutator subgroup of a group is less than the order of the group, there must be an invariant subgroup of prime index in the group, since the commutator quotient group is always abelian. While every simple group of composite order is evidently a perfect group, there are perfect groups which are composite. EXERCISES 1. The smallest perfect group which is not also simple is of order 120. 2. The factors of composition of the symmetric group of degree , n5^4, are 2 and !/2. 3. Everj' perfect group besides the identity is insolvable, but an insolv- able group is not necessarily perfect. 4. Every subgroup of a solvable group is solvable. 5. Each one of the series of successive commutator subgroups is invari- ant under the original group. 6. Every solvable group of composite order contains an invariant subgroup which is abelian and whose order exceeds unity. 69. Series of Composition. If each one of the series of groups (A) Go, Gi, G2i . . . , Gp = l, excluding the first, is a maximal invariant subgroup of the one which immediately precedes it, the series is said to be an ordinary series of composition of Go. For brevity an ordinary series of composition is often called a series of composition. A necessary and sufficient condition that Go be a simple group 178 SOLVABLE GROUPS (Ch.VIII whose order exceeds unity is that this series consists of only two terms. We may form another series (B) Go, G I, G 2, . , G\ = l in which each subgroup is an invariant subgroup of Go and has the property that no larger invariant subgroup of Go, con- taining this one, exists in the group which immediately pre- cedes it. The series {B) is said to be a chief series of composi- tion of Go. It is sometimes possible to select a series of com- position in such a manner that it is also a chief series. This can evidently always be done when the order of Go is a power of a prime number. It is always possible to construct an ordinary series of com- position by inserting some terms in a chief series of composi- tion if the chief series is not already an ordinary series of com- position. Suppose that it is necessary to insert some terms in the series (B) between G'^ and G'a+i to obtain an ordinary series of composition and that Hi is such a term, which corresponds to a maximal invariant subgroup in the quotient group G'a/G'a+i, while H2 is a conjugate of Hi under Go. Since Hi and H2 are maximal invariant subgroups of G' their cross- cut is also invariant under G'a, and the corresponding quotient group is the direct product of two conjugate simple groups. When Go is solvable these simple groups have the same prime order p, and hence this quotient group is of order p^. As every group of order p^ is abelian, it results that the cross-cut of Hi and any of its conjugates under Go involves the pth power of every operator in these conjugate subgroups as well as the commutators of all their operators. Hence it follows that if we find the complete set of conjugates of Hi under the solvable group Go, their common cross-cut, which is invariant under Go and hence coincides with G'a+i, involves all their commutators as well as the pth powers of all their opera- tors. This proves the following theorem: // Go is any solvable group and Go, G/, G'2, , G'\ = l is a chief series of composition, then the quotient group of any of the groups in this series with respect to the one immediately 69) SERIES OF COMPOSITION 179 following it is an ahelian group which involves only operators of prime order besides the identity. This abelian group is therefore of type (1, 1, 1, ... ). If the order of this quotient group is p", we must evidently insert a conjugates of Hi in order to obtain an ordinary series of composition from the given chief series. The first of these can be chosen in (/>" !)/(/> 1) different ways. When Go is insolvable, the given method of proof leads di- rectly to the results that the quotient group of any one of the groups, in the given chief series, with respect to the one immedi- ately following it, is a direct product of simple groups of com- posite order which are simply isomorphic. If these simply isomorphic simple groups are of prime order, we have the result expressed in the preceding theorem. It is easy to prove that the totality of the quotient groups of each group of a chief series of composition with respect to the one following it is an invariant of the group. In fact this proof is practically the same as the proof of the fact that the factors of composition of any group is an invariant of the group. The theorem that the quotient group of any group in a chief series of composition, with respect to the one which follows it, is a power of a simple group, results also directly from the fact that this quotient group cannot involve a characteristic sub- group. The given method of proof leads directly to the theorem that a necessary and sufficient condition that a group does not contain a characteristic subgroup is that this group is a power of a simple group. If this theorem had been assumed as known, the fact that each of the given quotient groups is a power of a simple group would not have required any proof. If we form the successive commutator subgroups of a solv- able group Go we obtain a third series (C) Go, G I, G 2, , G y = \. As the quotient group of each of these groups with respect to the one which immediately follows it is abelian and as each one of the successive commutator subgroups is invariant under Gt), it results that as regards the three series A, B, C we have 180 SOLVABLE GROUPS [Ch. VIII p>\>y. We may clearly pass from the successive commuta- tor subgroups series to a chief series of composition by insert- ing additional terms wherever necessary. It should be ob- served that in the successive commutator subgroups series all the terms of the series are invariants of Go, and hence the order of the quotient group of each term of the series with respect to the one just following it is also an invariant of Go, while in series A and B we proved only that the totality of these quo- tient groups is an invariant of Go, but the order in which these two quotient groups occur is not necessarily an invariant in these two series. To illustrate the difference between the series A, B, C, we first use for Go any group of order p'^, m>2. It is clear in this special case that p = \ = m, while 7<w/2 when m is even and 7<(w+l)/2 when m is odd. This results directly from the fact that every group of order p~ is abelian. If Go is the symmetric group of order 24 it results that p = 4 while X = -y = 3. A necessary and sufficient condition that Go be abelian is that 7 = 1. Although it is always possible to pass from a successive commutator subgroups series to a chief series of composition by inserting (if necessary) terms into the former series, it is not always possible to obtain the commutator sub- groups series by dropping terms out of a chief series of composi- tion. Similar remarks apply to series A and B\ that is, it is not always possible to obtain a chief series of composition by omitting terms of an ordinary series of composition, but it is always possible to obtain an ordinary series of composition by inserting terms, if necessary, between terms of a chief series of composition. The fact that it is not always possible to obtain a chief series of composition by omitting some groups of an ordinary series which is not also a chief series, should be emphasized, since erroneous statements in regard to this matter are rather common. For instance, such erroneous statements appear in the Encyclopedie des Sciences Malhematiques, tome 1, volume 1, page 568. As a very simple illustration it may be observed that if G is the octic group, and if Gi is a non-cyclic 70] INVOLVING ONE NON-CYCLIC 8YL0W SUBGROUP 181 group of order 4 contained in G, we. may construct an ordinary series of composition of G by using for G2 a subgroup of order 2 contained in Gi but not invariant under G. By omitting Gi from this series there results a series which is not a chief series of composition of G, since Gi contains the commutator subgroup of G. The fact that a commutator subgroup series of compo- sition cannot always be obtained by omitting subgroups from a chief series which is not also a commutator subgroup series can be illustrated by means of the direct product of the octic group and a group of order 2. A solvable group of composite order must be composite, but not every composite group is solvable. Sometimes the proof that all the groups which belong to a certain system are com- posite is equivalent to the proof that they are solvable. This is clearly the case when the invariant subgroups and the cor- responding quotient groups belong to the same system. For instance, the proof that every group whose order is the product of distinct prime numbers is composite is equivalent to the proof that all such groups are solvable. Similarly, the proof that every group whose order is a power of a prime is com- posite is equivalent to the proof that all such groups are solv- able. On the contrar}', the proof that every group whose order is divisible by 2 but not by 4 is composite does not estabUsh the fact that such a group is solvable. If it could be proved that every group of odd order is composite, it would result from this that every group whose order is not divisible by 4 would be solvable. 70. Groups Involving no More than one Non-cyclic Sylow Subgroup. One of the most useful theorems as regards solv- able groups is the one which afl5rms that a group is solvable if it involves either no non-cyclic Sylow subgroup or contains only cychc Sylow subgroups 'besides those whose orders are divisible by the highest prime which divides the order of the group. To prove this theorem we assume that the order of such a group G is written in the form g = pi'^p2'^ . />x"^, where p\, p2, - - , p\ ^rt distinct prime numbers, arranged in ascending order of magnitude. 182 SOLVABLE GROUPS [Ch. VIII Since G involves operators of orders />i"", the number of its operators whose orders divide g/pi is less than g. This number is known to be a multiple of g/pi ( 25) and hence it can be written in the form kg/ pi, where k is an integer. The number of operators of G whose orders are divisible by pi"^ is there- fore equal to g-kg/pi=l{pi-l), since this number is also a multiple of the number of the differ- ent possible generators of a cyclic subgroup of order pi"^. The first member of the given equation is divisible by g/pi, and, as each of the prime factors of this divisor exceeds pi I, it results that / is also divisible by g/pi. Hence k = l, and I=g/Pi- If ai> 1, it can be proved, in exactly the same way, that the number of the operators of G whose orders are divisible by />!" ^ but not by pi"' is g/pi-kig/pi^=liipi-l). Hence ^i = l, and li=g/pi^- By continuing this process it results that the number of operators of G whose orders divide Pfi'fi . P\">', /3<X, is exactly equal to this number. In par- ticular, G contains only one Sylow subgroup of order px'x, and the corresponding quotient group contains only one sub- group of order ^x_i"x-i, etc. Hence G contains a cyclic quo- tient group of order />i"', and the invariant subgroup of G which corresponds to the identity in this quotient group is such that each of its Sylow subgroups, with the possible exception of those of order />xx, is cyclic. This completes a proof of the following theorem: // all the Sylow subgroups whose orders are not divisible by the highest prime which divides the order of the group are cyclic, the group is solvable. In particular, every group whose order is not divisible by the square of a prime number is solvable. Hence there is only one such group when none of these primes diminished by unity is divisible by another. 711 SUCCESSIVE GROUPS OF INNER ISOMORPHISMS 183 71. Groups whose nth Group of Inner Isomorphisms is the Identity. Let h, /2, . . . , / represent a series of successive groups of inner isomorphisms of a group G. A necessary and sufficient condition that /i be simply isomorphic with G is that the central of G be the identity. In this case, all of the successive groups of inner isomorphisms are simply isomorphic with G. If / has the same order as la+i, a<n, then / is simply isomorphic with all the groups of the given series which succeed /. It may happen that for a sufficiently large value of n, / = !. In this case G is clearly solvable. Suppose that one of the prime numbers which divide the order of G does not divide the order of /. From the theorem that a group has exactly the same number of Sylow subgroups of any order as its group of inner isomorphisms has, where the identity is counted as a Sylow subgroup of order p" if the order of the group of inner isomorphisms is not divisible by p, it results that G involves only one Sylow subgroup of order />"* whenever the order of /, for a sufficiently large value of n, is not divisible by p. If G is a solvable group whose order is divisible by p, while the order of / is not divisible by p, then G is a direct product of its Sylow subgroups of order p"* and some other subgroup. On the other hand, it is evident that when G is such a direct product, then it is possible to find a number n such that the order of / is not divisible by p. Hence the theorem: A necessary and sufficient condition that a solvable group G be the direct product of a Sylow subgroup of order p^ and some other subgroup is that the order of the nth group of inner iso- morphisms of G should not ' be divisible by p when n is sufficiently large. In particular, a necessary and sufficient condition that G be the direct product of its Sylow subgroups is that we can arrive at the identity by forming successive groups of inner isomorphisms of G* * Cf. Loewy, Malhematische AnnaUn, vol. 55 (1901), p. 69. 184 SOLVABLE GROUPS [Ch. VIII EXERCISES 1. If the th succcssfve group of inner isomorphisms of a group is the identity, the wth successive group of inner isomorphisms of each of its subgroups must also be the identity. 2. A group of order 455 is necessarily cyclic. 3. The direct product of any finite number of solvable groups is again a solvable group. 4. If n is the number of operators or subgroups in a complete set of conjugates of a simple group, the order of this group divides n\ and is a multiple of n. 72. Arbitrary Choice of Factors of Composition. If a group G is the direct product of its Sylow subgroups, then each sub- group of G is also such a direct product, since we must arrive at the identity by forming successive groups of inner isomorphisms. Hence it results directly that it is possible to find a series of composition of G such that the factors of composition occur in an arbitrary order, whenever G is the direct product of its Sylow subgroups. On the other hand, suppose that G is a solvable group such that it is possible to select a series of composition in such a manner that all the factors of composition which are equal to a given prime number p appear at the end of the series, while all the other factors of composition are prime to this number. Hence G contains one and only one Sylow subgroup of order />". In particular, there results the theorem: A necessary and sufficient condition that a series of composi- tion of a solvable group G can be found which corresponds to an arbitrary arrangement of the factors of composition is that G is the direct product of its Sylow subgroups. By combining the theorem of the present section with those of the preceding one, it results that a necessary and sufficient condition that a series of composition of a solvable group can be found such that the corresponding factors of composition occur in an arbitrary order, is that we can arrive at the identity by forming successive groups of inner isomorphisms. These results can readily be extended so as to apply to groups which have composite factors of composition. Such a group is also the direct product of groups whose orders are powers of given 73] GROUPS OF ORDER ?>V 185 factors of composition whenever there is a series of comp)osi- tion corresponding to every possible order of the factors of com- position. 73. Groups of Order p" g'', p and q being Prime Numbers. In 138 it will be proved that every group of order p''(f is solvable. Two very simple special cases will be considered here. The case when /3 = 1 is especially simple and will be considered first. If G is of order p'^q and q<Pj then G contains only one subgroup of order />" and it must therefore be solvable. When q> p and G contains more than one subgroup of order />", it must contain just q such subgroups. We proceed to prove that no two of these subgroups can have a cross-cut whose order exceeds unity. Let K represent the largest possible cross-cut of a pair of these subgroups. Since K is invariant under operators of each of these subgroups which are not contained in K, it must be invariant under a subgroup of G whose order is divisible by some prime number besides p. Hence K is invariant under an operator of order q, and it is therefore contained in each one of the q subgroups of order p". Since K is composed of all the operators which are common to a complete set of conjugates, it is invariant under G, and the corresponding quotient group has an order which is of the same form as the order of G.. It remains therefore only to consider the case when the q sub- groups of order P" are such that no two of them have two operators in common. In this case these q subgroups are transformed according to a transitive substitution group of degree q which involves no substitution whose degree is less than ^1. It can therefore not contain more than ^1 substitutions of degree q, since the average number of letters in all its substitutions is qh Hence it contains only one subgroup of order g,,and it must therefore be composite. As no group of order p'q can be simple, every group whose order is of this form must be solvable. The case when all the Sylow subgroups of a group of order pq^ are abelian is almost equally elementary. Let G be such a group and suppose that G is simple. If s represents any 186 SOLVABLE GROUPS [Ch. VIU operator of order p contained in G and if J is any Sylow subgroup of order ^, it is evident that 5 can be transformed into all its conjugates under G by means of the operators of J. If each operator of G is represented by a substitution according to which this operator transforms the conjugates of s, there results a substitution group S which is simply isomorphic with G, since G is simple. To the subgroup 7 in G there corresponds a transitive sub- stitution group in 5, since J transforms s into all its conju- gates. As a transitive abelian group is regular, s has exactly c^ conjugates under G. Hence the identity is the cross-cut of any two of the (f Sylow subgroups of order p" contained in G. That is, if G were simple it would contain (/?" 1)^ opera- tors whose orders are powers of p, and hence it could contain only (f operators whose orders are powers of q. As such a group could contain only one subgroup of order ^, it could not be simple. This proves that every group of order p^cf is solv- able whenever the Sylow subgroups of orders p" and q^ are abelian. 74. Insolvable Groups of Low Composite Orders. From the theorems which have been established it follows directly that every group whose order is less than 60 is solvable. That there is an insolvable group of order 60 and that this is the lowest order of a simple group of composite order was observed by E. Galois. We proceed to prove that there is only one insolv- able group of this order. If a group of order 60 contains only one subgroup of order 5, the corresponding quotient group is of order 12 and hence the group is solvable. Hence an insolv- able group of order 60 must involve 6 conjugate subgroups of order 5 and must transform them according to a transitive sub- stitution group of degree 6 and order 60. As there is only one such substitution group,* there is only one insolvable group of order 60. * The fact that there could not be more than one such substitution group may be seen as follows: Such a group contains the group of degree 5 and of order 10, and hence it involves exactly 15 substitutions of order 2. As none of these can occur in two subgroups of order 4 the group must contain five subgroups of this order. 74] INSOLVABLE GROUPS OF LOW COMPOSITE ORDERS 187 The first order beyond 60 which is not included in the given theorems is 72. As this is of the form S/?"*, p being an odd prime number, we shall prove that no group whose order is of this form is insolvable. If such a group were insolvable it would contain 4 or 8 subgroups of order />"*, and hence p would be 3 or 7. In the former case the 4 subgroups of order f* would be permuted either according to the symmetric or according to the alternating group of degree 4. In either case there would be a solvable invariant subgroup and the corresponding quotient group would also be solvable. Hence the entire group would be solvable. If there were 8 subgroups of order p^, then p would be 7, and the 8 subgroups of order />"* would be permuted according to the group of degree 8 and of order 56. As this group is solvable, it results that no group whose order is of the form 8/>" can be insolvable. Similarly it may be observed that no group whose order is of form 4/>'" can be insolvable. The only remaining number less than 100 which requires consideration is 84. Every group of order 84 involves an invariant subgroup of order 7. Hence such a group is solvable, and there is one and only one insolvable group whose order is less than 100. As there is no number be- tween 99 and 120 which could be the order of an insolvable group we proceed to consider the insolvable groups of order 120. Since a group of order 120 involves either one or six sub- groups of order 5, every insolvable group of this order contains 6 subgroups of order 5, which it transforms according to a transitive group of degree 6. Every group of order 120 which contains six subgroups of order 5 is insolvable, since the order of the group according to which these six subgroups are transformed is either 60 or 120. If this order is 120 the group in question must be simply iso- morphic with it, and hence it must be simply isomorphic with the symmetric group of degree 5. If the order of this group of degree 6 is 60, the groups in question must have a (2, 1) isomorphism with the icosahedral group and must 188 SOLVABLE GROUPS [Cm Mil involve two substitutions (51, $2) which satisfy one of the following two sets of conditions: In the former case si , 52 generate the icosahedral group, and the group in question is the direct product of this group and the group of order 2. In the latter case si, S2 generate the group of order 120, which has no subgroup of order 60, but has the icosahedral group for a quotient group. This proves that there cannot be more than three insolvable groups of order 120. As it is well known that three such groups exist, it has been proved that there are only three insolvable groups of order 120; viz., the symmetric group of this order, the direct product of the icosahedral group and the group of order 2, and the group of order 120 which involves operators of order 4 and has the icosahedral group for a quotient group. Between 120 and 168 are four numbers which are not included in the general theorems which were established above. These numbers are 132, 140, 144, and 156. If a group of any of these orders were insolvable it would have to be simple. Hence we may confine ourselves to a proof of the theorem that every group of these orders must be composite. If a group of order 132 were simple, it would permute its twelve subgroups of order 11 according to a simply iso- morphic transitive substitution group of degree 12. As this would be of class 11 it would contain an invariant subgroup of order 12. A group of order 140 = 22- 5- 7 contains a characteristic subgroup of order 5, a group of order 144 contains 1, 3, or 9 subgroups of order 16, and hence cannot be simple, and a group of order 156 = 22- 3 -13 contains a characteristic subgroup of order 13. Hence 168 is the lowest order, beyond 120, of an insolvable group. We proceed to prove that there is only one simple group, and hence only one insolvable group, of this order. That there is at least one such group results from the theory of the transitive substitution groups of degree 7, and 74) INSOLVABLE GROUPS OF LOW COMPOSITE ORDERS 189 from the group of isomorphisms of the abelian group of order 8 and of type (1, 1, 1). A simple group of order 168 contains 8 subgroups of order 7, and can therefore be represented as a transitive substitution group G of degree 8. A maximal subgroup of G is of degree 7 and of order 21, and it therefore involves seven subgroups of the form (abc-def). Each of these subgroups is transformed into itself by six substitutions under G. Hence it may be assumed that all the possible simple groups of order 168 con- tain a particular subgroup of degree 7 and order 21, and are generated by this subgroup and a substitution of the form ab-cd-ef-gh, which transforms into itself a particular subgroup of the form (abc-def) contained in the given subgroup of order 21. If the given subgroup of order 6 is the symmetric group of this order, the three possible substitutions of the form ab-cd-ef-gh are completely determined by the subgroup of the form (abc-def). That is, there is not more than one transitive group of degree 8 which contains a particular sub- group of degree 7 and of order 21, and which is such that its six substitutions which transform into itself a particular subgroup of the form (abc-def) constitute the symmetric group of order 6. It is clear that there is one such group, since the simple group of degree 7 and of order 168 trans- forms its eight subgroups of order 7 according to a transitive group of degree 8. To prove that there is only one simple group of order 168 it remains only to prove that a transitive group of degree 8 and of order 168 cannot be simple if the subgroup of order 6 which transforms a subgroup of the form (abc-def) into itself is cyclic. In fact, such a transitive group contains 28 cyclic subgroups of order 6, and hence it contains 48 -|- 56-1-56 sub- stitutions of orders 7, 3, and 6 respectively. It can therefore contain only one subgroup of order 8. Hence we have com- pleted a proof of the theorem that iJtere is only one simple group of order 168, and hence there is also only one insolvable group of this order. 190 SOLVABLE GROUPS (Ch. VIII The only order between 168 and 200 which is not included in the general theorems which have been established is 180, and it is known that there is at least one insolvable group of this order, viz., the direct product of the icosahedral group and the group of order 3. We proceed to prove that this is the only insolvable group of order 180. A group of this order must have one, six, or thirty-six subgroups of order 5. If it has only one such subgroup it is evidently solvable. It could not contain thirty-six such subgroups, since these sub- groups would form a single set of conjugates. As an operator of order 2 could not have 45 conjugates, the group would contain operators of order 6. As the number of its operators of each of the orders 3 and 6 would be even and a multiple of 5, and as it would contain just 36 operators whose orders are prime to 5, it results that such a group would involve not more than 5 subgroups of order 3. Hence there is no group of order 180 which contains 36 subgroups of order 5. It remains to consider the possible groups of order 180 which contain exactly six subgroups of order 5. In this case the groups must be isomorphic with the icosahedral group on six letters and they must therefore involve an invariant subgroup of order 3. If we can prove that such a group G must also con- tain the icosahedral group invariantly, it must be the direct product of the icosahedral group and the group of order 3, and hence there is only one such group. If the subgroups of order 9 ' in G are non-cyclic, it is evident from the given isomorphism that G contains two operators, si, S2, which satisfy the conditions Si^ = S2^ = {siS2y = l, and hence G con- tains the icosahedral group invariantly. It remains there- fore only to prove that the subgroups of order 9 in G cannot be cyclic. If these subgroups were cyclic, G would clearly contain two operators /i, /2 which would satisfy the conditions /i^ = /2^ = {iihy, /i = l. Hence the equations ti =t2tlt2, t2 =tit2tl. 74) INSOLVABLE GROUPS OF LOW COMPOSITE ORDERS 191 If we consider the powers of the commutator we have (tit2trU2-^y=tr%t2^ti%%=ti-%t2Hit2% {ht2h -H2-^f=h- ^tlt2Hit2HiH2% =tr ^tit2%t2%t2Hl = h- Hit2HxH2hH2Hi = /i - Hit2Hxt2Hit2Hi =h-niH2hH2HiH2h^ = h-HiH2HiHi^h^ (hhh -U2-^y = tr ^HiH2HiH2ni^ ht2Hit2Hx = tx^ As txt2tx~^t2~^ is the product of t\t2 and /l-^/2~^ and as t\t2 and /2/1 have a common square, it results that tx^ is transformed into its inverse by /1/2. Since it is also invariant under /1/2, the order of tx cannot exceed 6; but this order divides 9 and hence it must be 3. That is, G involves the icosahedral group, and hence there is only one insolvahle group of order 180, viz.^ the direct product of the group of order 3 and the icosahedral group. We have now considered all the possible insolvable groups whose orders are not greater than 200 and found that there are six such groups, vjz., one of each of the orders 60, 168, 180, and three of order 120. From the simplicity of the above considerations it appears probable that these enumerations could readily be carried much further, but enough may have been done to exhibit the general nature of the problems involved. Several years ago Holder carried this investigation through all groups whose orders are less than 480 with the results exhibited in the following table: Insolvable Groups * Order 60 Number of groups. .... 1 120 168 180 240 300 336 360 420 3 1 1 8 1 3 6 1 Holder, Mathematische Annalen, vol. 46 (1895), p. 420. 192 SOLVABLE GROUPS [Ch. VIII EXERCISES 1. A group whose Sylow subgroups of order 2*" are cyclic contains an invariant subgroup of index 2. Suggestion: Represent the group as a regular group. 2. If a primitive substitution group of degree n is solvable, n must be a power of a prime and the primitive group must be contained in the holomorph of the abelian group of order n and of type (1,1,1, . . . ). Suggestion: Consider a chief series of composition of the primitive substitution group. 3. If a transitive group of the prime degree p is solvable, its order is a divisor of />(/> 1). I PART II* FINITE GROUPS OF LINEAR HOMOGENEOUS TRANS- FORMATIONS CHAPTER IX PRELIMINARY THEOREMS Linear Transformations, 75-82 75. Introduction and Definition. It is often of importance in analysis to exchange one set of variables for another, the variables of either set being linear homogeneous functions of the variables of the other set (cf. Ch. XVIII), as in coordinate geometry : x = x' cos Qy' sin Q. (1) y=%' sin Q-\-y' cos Q. We assume that a function /(:*;, y) is given, in which the new variables {x\ y') are to be put in place of the old (x, y) by means of (1) ; this is called operating upon f by the linear transforma- tion (l). A capital letter is in general used to denote a linear trans- formation; thus, we shall here denote (1) by S. The result of operating upon f{x, y) by S may then be indicated sym- bolically as follows: (2) (f)S=f{x' cos e-y' sin d, x' sin B-\-y' cos d). * This part was written by H. F. Blichfeldt. 193 194 LINEAR TRANSFORMATIONS [Ch. IX However, since/ is generally subjected to several transforma- tions successively, we find it convenient after each operation to drop the accents. In the future we shall therefore write, instead of (2), (f)S=f{x cos 6y sin 6, x sin d+y cos $). Formal Definition. A linear transformation in n vari- ables x\,X2, . . , Xnis a set of linear homogeneous equations Xi=aiix\-\-ai2x'2-\- . . . -\-ainx'ni A: Xn='an\X'\^-an2x'2+ +aT', expressing the original variables x\, . . . , x^ in terms of new variables x'l, . . . , x'n, under the condition that the equations can be solved for the latter. 76. Matrix and Determinant. It is customary to represent the transformation A above by the matrix of the coefficients (called the matrix of A): or simply an ai2 . . . ai *nn. The determinant of this matrix is called the determinant of A . 77. Inverse of a Linear Transformation. The solutions of the equations A ( 75) for a;'i, . . . ,:', say x'i=buXi-^h\2X2-\- . . . +&in^, x'2 = etc., etc., will constitute a new linear transformation which will take the conventional form adopted for A after the accented and unac- cented variables have been interchanged: Xi=bnx'\-\-bi2x'2-\- . . . -\-binx'n, B: Xn=bnlX\+bn2x'2-\- +&nn^'. This is called the inverse of A, and is denoted by ^4"^ (cf.;! 22). 78) PRODUCT OF LINEAR TRANSFORMATIONS 196 78. Product of Linear Transformations. If a function / be subjected to two linear transformations successively, A and B, the result is equivalent to operating upon / by a single linear transformation C, called the product of A and B. We shall prove Theorem 1. The proditct oj two linear transformations in the same variables, A=[a,t], B = [bJ, is a linear transformation C = [f,J, where n Csi = a,ibu-\-agJ)2t-{- . . . -\-asabt= x^flw^a; symbolically, AB = C. Proof. Consider a function f{xi, . . . , Xn) operated upon by^: (f)A=fiyi, . . . ,y), where ( 75) )>$= y ^asyXf,. The result of operating upon (f)A by B, namely {{f)A)B, which we shall write (f)AB, is then/(2i, . . . , z), where n n n ^ n . n Accordingly, if C be the linear transformation defined above, we have (f)C=fizi, . . . , Zn) = (f)AB. Note that the element c,i in the product -45 = C is obtained by multiplying the elements of the 5th row of A into the corre- sponding elements of the /th column of B and adding the results. 79. The Commutative and Associative Laws. The com- mutative law does not hold in general; that is, C=AB and C'=BA do not represent the same linear transformation. On the other hand, the associative law holds always: A{BC) {AB)C whatever be the transformations A, B and C; that is, if we write S for the product BC and T for AB, then AS = TC 196 LINEAR TRANSFORMATIONS [Ch.- IX This we prove by comparing the matrices of these products, as obtained by the application of Theorem 1. 80. Canonical Form of a Linear Transformation. Identical and Similarity-transformations. A transformation whose matrix has zero elements everywhere except in the principal diagonal is said to have the canonical form (or to be written in the canonical form) : Xl=a\X I, X2'=C12X 2, Xn ClnX n In such a case we employ the notation S = {<xi, a2, . . . , ai^. If the coefficients ai, a2, , an, which are called the multipliers of S, are all equal, we say that 5 is a similarity- transformation; if they are all equal to unity, 5 is the identical transformation or the identity. Denoting the latter by E and any transformation by ^, we have EA=AE=A. 81. Power and Order of a Linear Transformation. Since the associative law holds for a product, it follows that we may write A^ for AA, A^ for {AA)A, etc., and call these products the second, third, etc., powers of A. Moreover, denoting the inverse of -4" by A~^, we have A'*A~'' = A~'*A" = E, and A~'* = {A-^y. The index laws hold for positive and negative integral powers if we interpret A^ as E. Usually no power of a linear transformation A taken at random will be the identity. If, however, such a power exists, we say that A is oi finite order, and the lowest power of A which equals the identity is called the order oi A. EXERCISES 1. Prove that the determinant of the product of two transformations A and B is equal to the product of the determinants of A and B. Hence prove that the determinant of A is the reciprocal of that of ^4"^ 2. Find the inverse of S, 75. 3. Prove that if A = B= P q r s then B is the inverse of A . 4. Construct AB and BA, where s r B= -9 P} ps-qr=l, [S 83l LINEAR GROUP 197 5. Find the general form of a linear transformation in three variables which is commutative with 5= a a ,0 b, ay^b. 6. Prove that a similarity-transformation is commutative with any linear transformation in the same variables. 7. Prove that the multipliers of a transformation of finite order are roots of unity (cf. 116). 82. Remarks. For the purpose of avoiding a possible confusion on the part of the student of the terms " literal substitution " and " linear substitution," the term " Unear transformation " has here been adopted throughout. A different value for Cst from that given in 78 is obtained by inter- preting the linear transformations A and 5 in a different manner or by writing last in the product that transformation which operates first, as is the custom with functional operators. Thus, Klein, Jordan and Burn- side regard the variables in the left-hand members of A, 75, as the new, and those in the right-hand members as the old variables (the accents being placed accordingly or entirely absent), and therefore get Csj=2"_i0iw^to instead of the value given in 78; while Weber, attach- ing the same meaning to the linear transformations, inverts the order in the product, writing BA where the authors mentioned write AB. On the other hand, Frobenius and Schur, though writing ys for x's in the equations A, 75, interpret linear transformations in the same way and arrive at the same results as the author of this Part II. The latter has deviated from his customary notation in papers published on the subject to the extent of dropping accents of variables that were formerly sup- plied with them and vice versa. Groups of Linear Transformations, 83-87 83. Linear Group. If to is an imaginary cube root of unity, the six transformations 1 Ai = [0 Ij w2 ^4 = ,w 0, 1 A2 = > 1 0, V As = .0 w, a ^3 = .0 <^. 0) Ae = 2 0. 2 3 , T = "4 6' 1. ,0 2. 198 GROUPS OF LINEAR TRANSFORMATIONS (Ch. IX form a group of order 6 which is isomorphic with the sym- metric group in 3 letters (cf. 1). Here Ai is the identical transformation; the inverse of Az is A5, while A2, A^. and Aq are their own inverses. The transformations ^3 and As are both of order 3; and ^2, ^4, ^6 all of order 2. A set of g distinct linear transformations Si, . . . , Sg will form a group of order g if the conditions of 22 are all fulfilled. Such a group we shall simply call a linear group. 84. Collineations and Collineation-groups. It is often con- venient not to regard as essentially different two transforma- tions whose matrices can be obtained one from the other by multiplying all the elements of one by a constant factor, as, for instance, in the case of S = The two transformations are then said to represent the same collineation. In other words, a collineation is specified by the mutual ratios of the elements of the corresponding matrix, not by the actual values of these elements. In practice it is customary to affix a factor of proportionality to either the old or the new variables to distinguish a collineation from a linear transformation; the collineation represented by 5 or T above would thus be written pxi = 2a;'i -\-Zx'2, pX2 = X'2' If A and B are two distinct linear transformations which represent the same collineation, then BA~^=A-^B is a simi- larity-transformation. For, let the common ratio of the ele- ments of the matrix of B to the corresponding elements of the matrix of A be 6, then we immediately verify that B =AS=SA, where 5 = (0, 6, . . . , 6). If now a linear group G of order g be giVen, two cases may arise. Either no two distinct transformations of G will repre- sent the same collineation, or there will be some one set of, say /, transformations which all represent the same collinea- tion. In the former case G contains g distinct collineations; 841 COLLINEATION-GROUP 199 in the latter, the g transformations of G can be arranged into g/f sets, of / transformations each, furnishing g/f distinct col- lineations. For, let i4i, . . . , ^4/ be any set of transforma- tions representing the same collineation, then A2Ar^=S2, , AfAr^=Sf are similarity-transformations and are all distinct. Hence, if the group F of similarity-transformations Si, S2, . . . , S/ contained in G (cf. Ex. 1, 87) is of order /, we have /=/. On the other hand, if A is an arbitrary transformation in G, then the / distinct transformations -451, , ^5/ all repre- sent the same colUneation, so that/'=/. Hence /'=/. The sets of G can therefore be exhibited as follows: Si, S2, . . , Sf', ASi, AS2, . . , AS/; BSi, BS2, . . , BSf] To each line will correspond a single collineation. Moreover, if the product of a transformation from a set (a) and a trans- formation from a set (/3) fall in the set (7), then the product of any transformation from (a) and any transformation from (/3) will fall in (7), since the two products merely differ by a similarity-transformation. Accordingly, the group G is (/, 1)- isomorphic with an abstract group E of order h=g/J, namely the quotient-group G/F (13). Since a collineation in n variables can be interpreted as a projective transformation in space of 1 dimensions by using homogeneous coordinates, the abstract group H becomes a group of operators of order h, called the collineation-group corresponding to G. 85. An Example. Take the linear group G of order 8: .4i = 1 .0 1. > A2 = 1 -1 0. , ^3 = -1 - 1, ,A4- = ,1 -1 Bi = ^1 .0 - 1, ,B2- = 1 1 0, ,Bs = -1 . 1. > ^4 = 1 -1 0. 200 GROUPS OF LINEAR TRANSFORMATIONS (Ch. IX The group F of similarity-translormations is here of order 2: A I and ^43. The table of 84 will therefore consist of four lines as follows: Ai, As] ^1, A3\ A2A1, A2A3; ^2, ^4; BiAi, B1A3; Bu 53; B2A1, B2Ar, . B2, 54. The corresponding collineation-group, of order 4, can be written by selecting a representative from each of the four lines, as A\, A2, Bi, B2. Remark. It will be noticed that in this example neither the set Ai, A2, Bi, B2, nor any other set that may be selected from the four lines will form a linear group of order 4. 86. Groups of Linear Transformations of Determinant Unity. It follows from the manner in which the product AB oi two Hnear transformations A, B is formed (78) that the deter- minant Dab of ^5 is the product of the determinants of A and B : Dab = DaDb. Now let ,5 be a transformation belonging to a collineation- group G of order g in w variables: A, B, . . . , o, . . . , and Ds the determinant of S. A linear group isomorphic with G and whose transformations have unity for the value of their determinants may then be constructed in the following manner. Let be a solution of the equation (3) e'' = Ds-\ and let S\ be the similarity-transformation {d, 6, . . . , 6), the value of whose determinant is 6". Then S2 = SSi is of determinant unity. Since (3) has n solutions, there are n different transformations associated with 5 in this manner, say 52"', . . , S2^^^- Hence the following correspondence: G: A, ... 5, . . . G2: /1 2 , . . . , yi2 ,(1) . . , 5: .(a). 5 871 UNEAR FRACTIONAL GROUP 201 It is easy to verify that the set of transformations in the last line form a linear group (Go) which is (w, l)-isomorphic with the collineation-group G; in fact, if AB = C, and a, /3 arbitrary accents, then where 7 is fully determined. As an illustration, let G be the collineation-group ^ = (1, 1), B = We find ^i< = (l, l),^i^ = (-l, -1); 5i^" = (i ), 5i^^' = (-|, -h), and therefore G2: ^2"^ = (1, 1), -1 52"> = ^2<^> = (-i, -1); 1 ^2^2) = -1 Oj It may, however, be possible to find a subgroup of G2 of lower order whose corresponding collineation-group is Ukewise G, and whose transformations also have unity for the value of their determinants (cf. 110). This will be the case if G is a group of odd order in two variables ( 97). When in the future we mention a group of linear transformations of deter- minant unity corresponding to a given collineation-group or linear group, we shall mean such a group G3 of lowest possible order, having the same collineation-group as that given or as that corresponding to the given linear group. 87. Linear Fractional Group. When only the mutual ratios of the elements of the matrices are of importance and not their actual values, we may adopt another mode of representing the operators, namely by writing them in linear fractional form. Let a given linear transformation be A: x, = a,ix'i + +ax', (j = l, 2,-. . . ,if), 202 GROUPS OF LINEAR TRANSFORMATIONS (Ch. IX and let the ratios xx/xn, , ^n-iA be denoted by yi, . . . ,yn-\ respectively.' Then from A we get y,= S ; ; ^ 1 U=i, 2, . . . ,n-i). ai>'i+ -rann-iyn-i-ronn To a similarity-transformation here corresponds the identity ys = y', (5 = 1, 2, . . . , n-1), and we see that the linear fractional group is simply isomorphic with the corresponding collineation-group and may be regarded as its equivalent. EXERCISES 1. Prove that the similarity-transformations contained in a linear group G form a subgroup which is invariant in G. 2. Prove that the determinant of a linear transformation belonging to a finite group is a root of unity (cf. 116). 3. Among the determinants of the transformations of a group G of order g let there be one which is a root of unity whose index is the power of a prime p (116). Prove that if all those transformations be eliminated whose determinants contain as a factor a root whose index is the highest power of p occurring among such indices, then will the remaining trans- formations form an invariant subgroup of G of order g/p. In particiJar, prove that the transformations of determinant unity form an invariant subgroup of G. 4. Let T be one of the transformations of the group in the last exer- cise whose determinant contains a factor 6 of index />", and assume that p is relatively prime to the number of variables n. Then we can always find a root of unity, say ^, of index />", whose nth power is e~ ^ If now all the elements of the matrix of T be multiplied by ^, the determinant of the new transformation T' will no longer contain as a factor a root whose index is a power of p. , Dt= w. Here ^= w, and T'= wj Now prove that if all the transformations be modified in this manner we shall obtain a group G' which is isomorphic with G. Evidently, to a possible similarity-transformation (a, a, ... , a) of G whose multipliers a are roots of unity of index a power of p will correspond the identity (1, 1, ... , 1) of G'. In such a case therefore the order of G' will be that of G divided by a power of p. 5. Construct the group Gj corresponding to the linear group of order 8 given in 85. For instance, let T (o :)) 88l CHANGE OF VARIABLES 203 6, The group listed in 83 is of order 6. Show that the corresponding coUineat ion-group and group d are of orders 6 and 12 respectively. 7. Construct the collineation-group and group Gj corresponding to the linear fractional group of order 6: y=y', W, 1-y', l/(l-y'). (y'-l)/y', y7iy'-l). 88. Change of Variables. Before subjecting a given func- tion /(x, y) to a linear transformation or a group of linear trans- formations, we may introduce a different set of variables in the function and in the transformations. We shall examine in detail the important case where the new variables are linear homogeneous functions of the old; in other words, where the new value of / is obtained from the old by subjecting the latter to a linear transformation T which expresses the change of variables considered. To illustrate, let 5 be the transformation S: x = x' cos 6y' sin 6, y = x' sin d-\-y^ cos 6. We now suppose that new variables X, Y are introduced, where T: :r=i(X+F), y=l.(X-Y), and correspondingly r: x'=\{r-\-Y'), y=l.(r-FO, where P= 1. The function / becomes U)T-f{\{X-^Y), 1(X-F))=F(X, F), say, and the transformation S expressed in the new variables (Si) is found by solving for X, Y from the equations |(X+F) =^(r +F0 cos 0-l(X'-Y') sin d, l{X-Y)=l(r-hY') sin 0-\-l.(X'-Y') cos d. 204 GROUPS OF LINEAR TRANSFORMATIONS [Ch. IX We obtain Si: X^X'^', Y = Y'e-*r The function F may now be subjected to the transformation S\, producing (4) {F)S, = {(J)T)S,=^if)TSi. Obviously, the final result could equally well have been obtained by operating first upon f hy S before introducing the new variables; that is, the final expression in (4) is also obtained by introducing X', Y' in {f)S=f{x' cos d-y sin e, x' sin 0+/ cos Q) by means of T', giving ({f)s)r=(f)sr. Hence we have {f)TSi = (J)ST', so that, since / is an arbitrary function, TSi=sr, or Si = T-^ST'. As remarked in 75, we drop accents after operating by a linear transformation. Accordingly, our final formula b (5) Si = T-'ST. In the general case, the same symbolic result is obtained. Hence the Theorem 2. Let there be given a linear group G: A, B, . . . , and a change of variables in the form of a linear transformation T; then we obtain in the new variables a linear group Gi: T-^AT, T-^BT, .... These two groups are simply isomorphic (13, 24), and we shall write the latter symbolically T'^GT. CHARACTERISTIC EQUATION 205 89. Characteristic Equation. If we add ^ to each of the elements in the principal diagonal of the matrix of a linear transformation A = [a^] and equate the resulting determinant to zero, we have an equation in 6 which is called the character- istic equation of A: (6) ail 6 ai2 fl21 ^22 & a2n Onl On2 Onn-d = 0. Theorem 3. If T and A be linear transformations, the roots of the characteristic equation of A are the same as those of T~ M T. To prove this theorem, let us put T~^AT=B = [b,t], whose characteristic equation is (7) bii-d . bin bnl = 0. Regarding as a variable temporarily, we denote the transfor- mations whose matrices are the left-hand members of (6) and (7) by A-d and B-d. Then, since T" M T = B, and T-^ST = S, where 5 is the similarity-transformation (6, 6, . . . , 6), we may readily prove that T~^(Ad)T = B d. Hence, if the determinants of T,A d and B dhe denoted by p, q, r, we have (cf. Ex. 1, 81) p~^qp = r, so that q = r. Accordingly, the coefficients of the various powers of in ^ and r are equal, and the theorem follows. The sum of the characteristic roots of A is called the char- acteristic oi A. It is equal to the sum of the elements in the principal diagonal of A, namely aii+a22+ . -\-chut' EXERCISES 1. Prove that if S, 88, is a similarity-transformation, then Si=S. Prove also that if S and T both have the canonical form, then Sx=S. 2. Prove (5) in the case of two variables directly by multiplying out the right-hand member (cf. 77, 78). 3. Find the characteristic roots and characteristic of a transforma- tion written in canonical form; also of the transformation (1), 75. 206 GROUPS OF LINEAR TRANSFORMATIONS [Ch. IX 90. Transitive and Intransitive Groups. Consider a group G in four variables whose transformations all have the typical form b A = a I g c h d If in this group we introduce new in variables yi, y2, 2i, Z2 such that yi=xi-{-X2, y2 = X3-\-X4; zi=xiX2, Z2 = X3X4, the trans- formations take the simpler form Ai = It seems natural to adopt the notation {A' 0' p q r s / u V w Ai = A" A' representing a matrix in yi, y2 only, and A" one in zi, Z2 only. Correspondingly we write IG' 0^ Gi = [0 G"] We say that the group G (or Gi) is intransitive, and that (yi, y2) and (zi, Z2) form its sets of intransitivity. Similarly, if in a group in four variables, these fall into two sets of 3 and 1 variables respectively, by a suitable change to new variables, the group is intransitive. In general, a group G in which the variables fall into two or more sets of intransitivity [G' 0' G = G"j 91] HERMITIAN FORM 207 either directly or after a suitable choice oj new variables shall be said to be intransitive. If such a division is not possible we say that G is transitive. EXERCISES 1. The group of order 6 listed in 83 is transitive. It contains an intransitive subgroup of order 3 and three intransitive subgroups of order 2 each. 2. Find the largest intransitive subgroup contained in the group of order 8 listed in 85. 3. Prove that if a group in four variables appears as intransitive by two different changes of variables, 5~ 'G5 and T~ ^GT, such that the sets of intransitivity in 5~ 'G5 contain (2, 2) variables, and in T' ^GT contain (3, 1) variables, then a change of variables V' ^GV can be foimd such that there will appear at least three sets of intransitivity (2, 1, 1). Note. Maschke introduced the term transitive as applied to linear groups {Mathematische Annalen, Bd. 52 (1899), p. 363). Jordan denoted both intransitive and imprimitive groups (106) by decomposable groups (Alti delta Rede Accademia delta Scienze fisiche e matematiche, Napoli, t. 8 (1879), p. 6). Hermitian Invariant, 91-93 91. Hermitian Form. If the conjugate-imaginary of a quantity w be written w, a positive-definite Hermitian form (or simply Hermitian form) is an expression such as subject to the conditions that it vanishes only if ^i=a;2= . . . =Xn = 0, and is real and positive for all other sets of values assigned to these variables. Theorem 4. A positive-definite Hermitian form J in n vari- ables may be reduced to the form yiyi-\-y2y2-\- . . . -i-ymym 208 HERMITIAN INVARIANT [Cu. IX by a change of variables of the following type: y2 = P2\Xl-\- P22X2, yi= P\X\ + ps2X2-\- . . . -^pnXt, yn= Pn\Xl-\- Pn2X2-\- +P^+ + PnnOCn. Proof. Arranging / according to ocn and Xn we have J == J n '^(InnXnXn'irXn-^n X \Xn-^n-\ 1 -^ > where Xn-\ represents a linear function oi x\, . . . , Xn-u The coefficient qnn is real and positive, since it is the value of J obtained by putting Xn = l, Xn-i=x-2= =ici = 0. Accordingly, and we may write \ Vg/\ y/qj qnn = ynyn-\-Jn-l: Say, where yn = ^qnnXn H -=, vg and is therefore a linear function of a^n, . . . , xi. The function Jn-i fulfils the conditions of an Hermitian form in n 1 variables ocn-i, . . . , Xi, as it is of the required tj'pe and is real and positive for any set of values allotted to these 1 variables except 0, . . . , 0. For, it is the value of Jn obtained by putting Xn= Xn-i/qnn- Hence, we may arrange Ja-i according to Xn-i and ic-i and proceed as above. We find Ju-l =yn-iyn- 1 -\-Jn-2, 92] CONJUGATE-IMAGINARY GROUPS 209 where yn-i is a linear function of x-i, x-2, . . . , Xi. Con- tinuing thus, we finally prove the theorem. 92. Conjugate-imaginary Groups and Invariant Hermitian Form. If in a group G we replace the variables xi, . . . , Xn and the elements a^ of the matrices by their conjugate- imaginary values xi, . . . , Xn, a,t, we evidently obtain a group G simply isomorphic with G. We shall say that either group is the conjugate-imaginary of the other. We say that an Hermitian form J is invariant under a group G, or that / is an Hermitian invariant of G, when / is trans- formed into itself by the (intransitive) group in 2 variables xi, . . . y Xn, xi, . . . , Xn made up of G and G. Theorem 5. There is always an invariant Hermitian form of a given linear group G in n variables * Proof. Let the transformations of the group made up of G and G be denoted by Ti, T2, . . . , Tg, and let / represent the Hermitian form xixi-fa;2a;2+ . . -\-XnXn. Then the sum j={i)Ti-\-(i)T2-\- . . . +(/)r is an invariant Hermitian form of G. First, / is an Hermitian form. For, each of the terms {I)Ta is the sum of n expressions {xsXt)Ta = XtXs which are real and non-negative. The function / is therefore real and non-negative, and cannot vanish unless every term {I)T vanishes. But, if Ti represents the identity, {I)T\=I and does not vanish unless every variable xi, . . . , Xn vanishes. This is therefore also the case with /. Second, / is transformed into itself by Ti, r2, . . . , Tg. For, evidently (7)r=((/)ri)r+ . . . +((/)r,)r=(/)ri+ . . . -f(/)n, where rfi = TffT^. *This theorem was proved for =3 by Picard and Valentlncr (1887, 1889), and for any n by Fuchs. Moore and Loewy (189(i). Sec Encyklopddie der Maihe- malischen Wissenschaflen, Leipzig, 1898-1904, Bd. I, 1; p. 532. 210 HERMITIAN INVARIANT (Ch. IX But, TiTft, . . . , ToTa3.re the transformations Ti, . . . , T, over again in some order. It follows that (j)r=(/)ri+ . . . -\-{i)T,=j. From Theorems 4 and 5 we get the Corollary. Such variables Xi, . . . , Xn may be selected for a group G that the function I=XiXi-j-X2X2-^ . . . +3cJcm is an Hermitian invariant of G. 93. Linear Transformations in Unitary Form. The variables of G being chosen as specified in the previous corollary, let -4=[a], ^=[a] represent corresponding transformations of G and G. Operat- ing upon / by -4 and A, we find the following conditions that / may be reproduced: aitau+a2ta2t+ . . . +atat = l (^ = 1, 2, . . . , w), aitfiii-{-a2ta2i-\r . -\-aniflni=0 {k,l = l,2, . . .,n;k9^l). The transformation A fulfilling these conditions is said to have the unitary form, or to be a unitary transformation. The inverse of A can here be written down at once: A~^ = [a'st] {a'u = au). For, the condition A~^A = \ht identity leads to the equations (8). Now, since A A ^ = the identity also, we obtain the follow- ing set of equations as consequences of (8) : atiati+at2ak2-\- . . +atnatn = l (^ = 1, 2, ... , ), atian-\-at2ai2-\-. . . +atuain = {k,l = l,2,. . . ,n;k9^l). 94. Reducible and Irreducible Groups. A group G is said to be reducible if, by a suitable choice of variables, it can be written in the symbolic form (cf . 90) : fC G = 94] REDUCIBLE AND IRREDUCIBLE GROUPS 211 that is, if a certain number of the n variables, say xi, . . . ,Xm, where m<n, are transformed into linear functions of them- selves by every transformation of G. For instance, a group in two variables is reducible if (either directly or after a proper change of variables) all the matrices are of type 'a 0^ b c If this is not the case, the group is said to be irreducible. We shall say that the m variables xi, . . . , Xm form a reduced set for G. Theorem 6. A reducible group G is intransitive, and a reduced set constitutes one of the sets of intransitivity of G. Applied to the illustration above, the theorem asserts that the group there given can be written in the form a o' by a suitable choice of variables. Proof. The group G has an Hermitian invariant, which by the change of variables specified in 91 may be written: yiyi-\-y^2-\- . +ynyn. Making the corresponding changes in G, this group is still seen to be of the form IG' G" G' namely, (10) ^*='""'/ + yt=(h\y 1+ . +<Im/m (5 = 1, 2, ... , fn), . \-(hmy'm+ . . . -\-atny'n (/ = w4-l, W4-2, ...,). Applying the conditions (8) and (9), 93, and writing Cm for the product a^a, we obtain, among others, the following 2(n m) equations: c+i.+ . . . +<:,= 1 (t) = w+l, m+2, . . . . n), Ci + . . . +<:. = ! (u' = m+l, m4-2, . . . , n). 212 CANONICAL FORM fCn. IX. If we now subtract from the sum of the last nm equations the sum of the first n-m, we get n _ m *-m+l (-1 The quantities Cu being real and non-negative, it follows that those which enter into this sum all vanish. Moreover, since a,t = follows from ^^ = 0, the equations (10) now take the form v.=a.iyi4- . . . +asmy'm is = l, 2, . . . ,m), yt=atm+iy'm+i-\- . . . +atny'n {t=m+l, w+2, ...,), and the theorem is proved. EXERCISES 1. Prove that if 'p q A = r s is a unitary transformation, then r= q and s=p. 2. Prove that if all the elements of the matrices of the transformations of G are real, then there is a quadratic function of the variables which is invariant under G. 95. Theorem 7. A linear transformation of finite order will assume the canonical form ( 80) hy a suitable choice of variables. Proof. Let the transformation be n A: Xs=^ ^ astx't (5 = 1, 2, ... , n). t-i Then we can always find a linear function yi = biXi + . . . -\-bnXn which is transformed into a constant (d) times itself by A (i.e., yi is a relative invariant oi A). For, we get n n n n (yi)A = 2L,^' Z^astx,= 2^x, 7^Mi, -i 1-1 (-1 -i and this expression is 6yi provided the following equations are true: n eb,= ^b,a (/ = 1,'2, . . . , ). -i S 96: ABELIAN GROUPS 213 By tJie theory of linear homogeneous equations, a set of solutions bi, . . . , bn, not all zero, of these equations, can always be found if is a root of the characteristic equation of^ (89). If we now introduce new variables such that yi is one of these, the group generated by A is reducible, since (yi)A=eyi, (yi)A^ = ef^yi, etC, and therefore yi = ey\ is one of the equations specifying A in its new form. By Theorem 6, the group generated by A is intransitive, one of the sets of intransitivity being yi. Let (y2, . . . , yn) form the other (temporary) set of intransitivity. The above process may now be repeated for the set (y2, . . . , yn)- We determine the linear function C2y2 + . . . +cy which is a relative invariant of A, and introduce this function as one of n 1 new variables to take the place of y2 , . . . , y. Continuing thus, the transformation A will finally appear in the canonical form. 96. Theorem 8. In any given ahelian group K ( 26) of linear transformations, such new variables may be introduced that all the transformations of K will simultaneously have the canonical form. Proof. If the group contains only similarity-transforma- tions, the theorem is self-evident. Hence we assume in K a transformation S which is not a similarity-transformation. Let the variables of the group be chosen such that 5 appears in the canonical form 5'f=(ai, . . . , ai; a2, . . i , a2; . . . ; a, . . . , a.), the variables being arranged so that those having the same multipliers are grouped together. Let there be a variables x\, . . . , Xa having the multiplier a\\ b variables Xo+i, . , Xa+b having the multiplier a2, etc. Now let T be any transformation in K. Since TS=STf 214 CANONICAL FORM [Ch. IX we find by applying the rule for forming products that T has the form Xt = a,\x'\-\- . . . -\-a,ax'a (5 = 1, 2, ... , a), Xt = ata+ix'a-^i-\- . . +ata+e^'o+ft (/ = a+l, a+2, . . . ,a+6), Hence we infer that K is intransitive, and if we now confine our attention to one of the sets of intransitivity, we may apply the above process to that set. This will, therefore, break up into further sets of intransitivity. Continuing thus, the ulti- mate sets of intransitivity contain one variable each, and the theorem is proved. Instead of the phrase " let the variables be so chosen that a (given) transformation (or group) will appear in the canonical form " we shall often say simply: " let the (given) transformation (or group) be written in canonical form." EXERCISES 1. Can the transformation 1 1 1 be reduced to the canonical form? Find the condition that the trans- formation a which is not necessarily of finite order, can be written in canonical form. 2. Prove that an abelian group in two variables can be written in canonical form and that at the same time the Hermitian invariant becomes CHAPTER X THE LINEAR GROUPS IN TWO VARIABLES 97. Introduction. We shall limit ourselves to the deter- mination of the groups whose transformations have unity for the value of their determinants. From these all other forms of groups may readily be constructed (cf. 83-87). We shall say that a given group is a type of all groups which may be obtained from it by a mere change of variables. All the types we encounter (with one exception) contain a group of similarity-transformations of order 2, E = {\, 1), E\ = { \, 1), due to the fact that a linear transformation of determinant unity and of order 2 does not exist unless it be the similarity-transformation Ei. A transformation whose corresponding collineation is of order 2, if written in canonical form, must necessarily be {i, i), where P=l. The excep- tion mentioned is the type of an abelian group of odd order. Following Jordan, it shall be our practice to call the order of a linear group g(t>, if g is the order of the corresponding collinea- tion-group, and (f> the order of the subgroup of similarity- trans- formations contained in the given group. There are several processes available for the determination of the types of groups sought.* We shall here employ a modi- fied form of Klein's original process, which depends largely on geometrical intuition. Klein, Mathematische Annden, Bd. 9 (1876), p. 183 ff.; Vorlesungen iiber das Ikosaeder, Leipzig, 1884, pp. 11&-120. Goidam, MalhenuUische AnnaJen, Bd. 12 (1877), p. 23 ff. Jordan, Journal fur die reine und angewandte Mathemaiik, Bd. 84 (1878), pp. 93-112; Atti della Reale Academia di Napoli, t. 8 (1879). Fuchs, Journal fiir die reine und angewandte Mathematik, Bd. 81, 85 (1876, 1878), pp. 97, 1 ff. Valentiner, De endelige Transformaiions-gruppers Theori, Copen- hagen, 1889, p. 100 ff. 216 216 THE LINEAR GROUPS IN TWO VARIABLES [Ch. X 98. Outline of the Process. 1. Let G be a group in two variables xi, X2- _Then by the introduction of the conjugate- imaginary group G (cf. 92) and by the selection of new vari- ables X, Y, Z which are bihnear in xi, X2 and their conjugate- imaginary values ^1, X2, we obtain a group G' of real rotations in space, leaving the origin fixed (99). 2. Consider now a sphere 2 of radius 1 whose center is the origin. With each rotation of G' belongs an axis of rotation. One of the points where such an axis pierces 2 together with all those points into which this point is moved by G' form the vertices of a regular polyhedron, including the limiting cases where there is a single axis of rotation or where the polyhedron becomes a flat polygon ( 100). 3. The determination of G' is therefore made to depend upon the construction of the analytical expressions represent- ing the rotations of the regular solids. We find five different types for G' and correspondingly five different types for the linear groups G ( 101-103). 99. The Group of Rotations G'. Let a b S = c d be any transformation of G, whose variables xi, X2 are chosen such that the Hermitian invariant is/ = a:iici+a;2iC2 (Cor., 92). Then the following equations are true (93, 97): ad hc = \=ad he, aa-j-bb = l, ac-{-bd = 0, cc-\-dd = l. From these we obtain c=b, d = a, Moreover_^ if we let p, q represent the positive square roots of aa and bb respectively, and put a = pa, b = q^, we get, since P=P> l^r- C=-q^, d = pa; 991 THE GROUP OF ROTATIONS G' Furthermore, if we put 7 = VajS and 5 = Va/zS, we have 77 = 55 = 1. Then it follows by direct multiplication ( 78) that S=S\S2Sz, 217 where Sx = '7 P q '5 0' , S2 = , ^3 = .0 7. -q P. .0 "5. The corresponding transformation 5 of the conjugate- imaginary group G can similarly be written as a product SiS2S'd, where 7 0- , 52 = P q , s,= "5 0' .0 7. -q P. .0 5, In these expressions we shall finally put 7 = cos tt f sin , S = coswisinw, p = cosv, u, V, w being real angles. We now introduce the new variables q = smv, X = xiit:i a:2a;2, Y =xiX2-\-X2Xi, Z=i{x\X2X2X\) These are transformed into linear functions of themselves by 5i, 52, 53, operating simultaneously with S\, S2, S3. In fact, we find (X)SiSi =X, {Y)SiSi = Y cos 2u-Z sin 2u, {Z)SiSi = Y sin 2m +Z cos 2u; or, to follow our previous practice, 10 5i5i= cos2w -sin2tt sin 2w cos 2k 218 THE LINEAR GROUPS IN TWO VARIABLES (Ch X Similarly we find cos 2v sin 2v S2S2 sin 2^ cos 2v 5353 = COS 2w sin 2w sin 2w cos 2w If we interpret X, Y, Z as rectangular coordinates in ordi- nary space, we recognize here three real rotations around the X-, Z-, X-axes respectively, the origin remaining fixed. The rotations performed successively will, as is well known, be equivalent to a single rotation. With the transformations of the group G are therefore associated rotations which evi- dently form a group G' isomorphic with G. The isomorphism is (1, 2) in the case where G contains i = ( 1, 1); other- wise it is (1, 1), since we may readily prove that to identity of G' will correspond only E = {1, 1) or Ei of G. In other words, G' is simply isomorphic with the collineation-group correspond- ing to G. 100. The Regular Polyhedron. Consider an axis of rota- tion (L) of G', and let the various angles of rotations around L be the different multiples of (360/w); we shall say that L is of index m. Let Pi be one of the points where L cuts the sphere 2. This point will be transformed into (say) k distinct points upon 2 by G': Pi, P2, . . . , Pt, all of which will be extremities of axes of rotation of index m. The distribution of these points about any one of them is similar to the distri- bution about any other. Now let arcs of great circles be drawn connecting Pi with all the other points P2, . . . , Pt, and let the shortest arc be of length A. The number of arcs of this length radiating from Pi is w or a multiple of m, since always m of the arcs are interchanged by rotations about L through the different multi- ples of (360/ w). However, there cannot be more than 5 arcs A ; an exception occurring where we have just one or two points Pi, P2, one or both extremities of L, in which case A = 360 or 180. For, if there were 6 or more, a pair of them (say C, C^ would make an angle /3^60 with each other at Pi; lOOl THE REGULAR POLYHEDRON 219 and, this being the case, the arc B connecting P, and P, (the points of Pi, . . . , Pt located on C, and d) would have a length <A. For, by trigonometry, cos B = cos^ A +sin2 A cos /3 ^ cos^ A -\-\ sin^ i4 > cos -4 ; and, since 0<5<90, it follows that B<A. But this is con- trary to hypotheses, since the lengths of the arcs radiating from P., are equal to the lengths of the arcs radiating from Pi. Let m>2. Then it follows that there are just m arcs of length A radiating from Pi, each making an angle of (360/m) with its adjacent arcs. The same will be true for each of the points P2, . . . , Pt, and we see readily that the sphere will be divided by all the arcs of length A , joining the various points Pi, . . . , Pt which can be reached from one of them by pass- ing along such arcs, into a number of equal and regular polygons. Accordingly, these points, say Pi, . . . , Pi, are the vertices of a regular polyhedron inscribed in S. Consider next the case where there are no axes of index greater than 2. Proceeding as above, we let L denote an axis of index 2, and we obtain the points Pi, . . . , Pt by G'. There are at least two arcs of length A radiating from Pi making an angle of 180 with each other. Taken together they form a single arc C upon which (when extended round the sphere) Pi and some other points P2, . . . , Pj lie, equally distributed over the entire circle. If / > 2, a rotation of 180 around Pi followed by a rotation of 180 around one of the points next to Pi is equivalent to a rotation of (720//) around an axis perpendicular to the plane of the circle C Every axis is of index 2 by assumption. It follows that 2// = l or 1/2; i.e., / = 4, In this case we have three mutu- ally perpendicular axes of index 2. The distance A is therefore either 180 or 90. In the for mer case we have a single axis of index 2 in G'. In the latter case there are four arcs of length A radiating from Pi, lying on two circles which are at right angles to each other at Pi. Their extremities lie in the diametral plane which is perpen- dicular to the axis L, and must be 90 or 180 apart. Con- 220 THE LINEAK GROUPS IN TWO VARIABLES [Ch. X sequently, G' contains just one axis of index 2, or just three such which are mutually perpendicular. The Groups of the Regular Polyhedra, 101-103 101. Limiting Cases. We notice first that the most general linear homogeneous change of variables {xi, X2) in G is indicated by a linear transformation T ( 88) to which again corresponds the most general rotation of 2 about its center. It follows that any given configuration arrived at in 100 may at the outset be placed in any required position relative to the axes of coordinates X, Y, Z. Beginning then with the simplest case where there is a single axis L of rotation, we let this be the X-axis. Then sin2z) = and cos22; = l (cf. 99). Hence S has the form (a, dba"^). If S is of order g we have (0:)'' = 1. (A) G': a single axis of index g; G: an abelian group (intransitive) of order g: 5x = (e\ 6-^); X=l, 2, . . . , g; e'' = l. The next case to be considered is where there is an axis L of index g, assumed to be the X-axis as above, in addition to g axes of index 2 lying in a plane perpendicular to L. Let one of the latter be the Z-axis; we then have cos 2z;= 1, cos 2{uw) = 1, and the corresponding transformation of G is found to be T= 1 =F1 (B) Dihedral Group. G' : one axis L of index g and g axes of index 2; G: 'an imprimitive group of order 2^0 consist- ing of the transformations 5x=fdze\ e-^), n= 102) THE TETRAHEDRON AND OCTAHEDRON 221 102. The Tetrahedron and Octahedron. We now examine the five ordinary regular solids. Of these, the hexahedron and octahedron furnish the same set of axes of rotation, as do also the dodecahedron and icosahedron. We therefore have only three cases to consider: the tetrahedron, octahedron and icosa- hedron. In the case of the tetrahedron we have four vertices and correspondingly four axes of rotation of index 3; besides, three axes of index 2, each passing through the middle points of a pair of opposite edges. The latter axes are mutually perpen- dicular and may be taken as the X-, Y-, and Z-axes. The corresponding transformations of G are then as follows: Ti = ii, -i), T2 = '0 i , T,= ' 1 [i OJ [-1 oJ either directly or after multiplication by jEi = ( 1, 1). If the vertices are named a, b, c, d, the three rotations permute them among themselves according to the substitutions (ab){cd), (ad) (be), (ac){bd). The remaining rotations permute the vertices three at a time cyclically, as (abc), . . . The corresponding transformations of G may be determined analytically from the conditions that they are each of order 3 and transform the coUineations corre- sponding to Ti, T2, T3 cyclically. Certain ambiguities arise from the fact that the similarity-transformation JSi = ( 1, 1) is present in the group. Thus, S = (abc) may transform Ti into T2 or into T2Ei, etc. For (abc) we find four forms possible, all of which are present in G if one of them is. We shall choose the following form : -l+i -!+ S = 2 2 1-f-t -l-i 222 THE LINEAR GROUPS IN TWO VARIABLES [Ch. X (C) Tetrahedral Group. C : generated by 5 = (abc) and Ti = (ab) (cd) ; G: a primitive group ( 106) of order 12<^ gen- erated by the transformations S and Ti above. The rotations of the octahedron include those of the tetra- hedron S = {abc), Ti = {ab){cd) if here a, b, c, d represent each a pair of opposite faces. To the list of generating rotations we now add one, U say, of order 4, having the same axis as T\, and lP = Ti, or U = (acbd). The corresponding trans- formation of G is readily found to be VV2 V2/ (D) Octahedral Group. G' : generated by 5 = (abc) and U= (acbd) ; G: a primitive group of order 24<f>, generated by 5 and U above. 103. The Icosahedron. An icosahedron contains 10 axes of rotation of index 3. These (counted twice) may be grouped in 5 sets of 4 axes each, such that the axes of each set are arranged in the same way as the axes of index 3 in (C). In this manner we obtain 5 regular tetrahedrons, each correspond- ing to a subgroup of order 12. The group is accordingly iso- morphic with the alternating group of order 60 and is generated by the substitutions S, Ti of (C) and a substitution V = {ab){de), whose corresponding transformation may be determined from the relations F2= or El, {TiVf = E or Ei, {SVy=E or i. The first and last ambiguities fall away, as of necessity V^ = {SV)^=Ei (cf. 97); and by using VEi if necessary in place of V we may take (TiV)^=E. We then find i ^-'^ v= 103] where THE ICOSAHEDRON 223 ^ I-V5 I+V5 (E) Icosahedral Group. G': generated by S = {abc), T\ = {ah){cd) and V^{ah){de)\ G: a primitive group of order 600 generated by S, Ti and V above. EXERCISES 1. Construct the analytical forms of the rotations of G' corresponding to the generating transformations Ti, Ti, Tz, S, U and V. Prove that the X-, Y-, Z-axes are permuted among themselves by all of these rotations except V, and hence that the group G' in the cases (C), (D), as a group in three variables, is imprimitive (cf. 106). 2. Determine a set of generators of (E) corresponding to the substi- tutions S'= {abode), U'={ad){bc), T'={ab){cd) of the alternating group in five letters; under the condition that S' is written in canonical form: 5'=(*',.*);*'=1. Hints: We first determine U' from the condition 1}'-^S'U'=S'-^: U'= p Pq=-l. The change of variables x=pX, y=Y will leave S' unaltered and will reduce U' to the form above, except that now P=l, q= l. We finally assume 'a /3" r= y 5 aS-0y=l. The condition T'^=Ei (97) gives us a + S = (t)+(-t)=0 (89), which, combined with f/'-^rf7'=r or r, (only T'E, will be compatible with the previous results) is equivalent to 5= a, 7 = 0; a*+/3*= 1. Finally, we have the relation (abcde)-(ab)(cd) = (bde); that is, the transformation S'T' is of order 3 and its characteristic roots are consequently w, w* or w, w*. The latter possibility can be avoided by taking T'Ei instead of T'. Applying this condition we finally have ""Vf' ^"Vfi- 224 THE LINEAR GROUPS IN TWO VARIABLES (Ch. X Invariants of the- Linear Groups in Two Variables, 104-105 104. General Theory. A homogeneous function of the vari- ables xi, X2 of a group G: Si, 02, , Sff^ is called an invariant of G (or we say that G leaves f invariant) when/ is transformed into a constant multiple of itself by G: (f)Sj = c^. Let / be resolved into linear factors. These are permuted among themselves by G, and the product of a set of them which are permuted transitively will evidently furnish an invariant by itself. This invariant, say F=fif2 ...//,, can readily be constructed by operating upon one of the factors /i by the transformations of G, and we shall call it a fundamental invari- ant. Any invariant is accordingly a product of fundamental invariants. If /i be selected at random, the corresponding fimdamental invariant is evidently of degree g. To obtain fundamental invariants of lower degree we make use of a theorem of transi- tive substitution groups, namely that the ratio g<f)/h is the order of that subgroup of G which leaves /i invariant. Now this subgroup, Gi, must be abelian. For we may change the variables, introducing /i as one of the new variables, say xi. Then Gi must appear in the form of a reducible group: a 0' and can accordingly be written as an intransitive group a 0' But this is the canonical form of an abelian group. It follows furthermore that two subgroups, d and G2, having in common a linear invariant /i, generate an abelian group. 105] THE FUNDAMENTAL INVARIANTS 225 The other factors /2, . . , /a of F are linear invariants of subgroups G2, . , Gh of G, conjugate to Gi. For, if Sa~^GiSa = G2, then (fi)Sa=f2 belongs to F and is an invariant of G2. Hence our problem becomes one of determining the different conjugate sets in G of abelian subgroups which are not subgroups in larger abelian subgroups. The fundamental invariant F will be made up of one factor for each of the subgroups of the set if there is no transformation in G which transforms one of the linear invariants of Gi into the other. In other words, if Gi be written in the canonical form (a, c), and if there is no transformation in G of type />' [q 0. we get two fundamental invariants by starting with /i = Xi and/i =X2; otherwise we get just one invariant, containing both xi and X2 as factors. 105. List of the Fundamental Invariants. We shall, of course, hmit ourselves to the invariants of degree <g. (A) Case (A), 101. Two invariants, Xi and X2. (B) Dihedral Group. One invariant, iCiX2. (C) Tetrahedral Group. There are two conjugate sets of sub- groups, of orders 20 and 30. The first set consists of groups conjugate to that generated by Ti, and the invariant is * t=XiX2(Xi* X2'^). The second set contains the groups conjugate to that gen- erated by S. Here we have two invariants, each containing just one of the linear invariants of S: ^=Xl^ -\-2V^Xi^X2^^-X2*; ^ = OJi" - 2 V^iCi%2Ha:2*. These invariants satisfy the relation 12V^/2_,j>3_|_^3^0. (D) Octahedral Group. Here we have three conjugate sets of subgroups, of orders 40, 30 and 20. In the first set The notation is that given by Klein, Vorlesungm, etc., pp. 51^58. 226 THE LINEAR GROUPS IN TWO VARIABLES [Ch. X there is a group generated by C/, and the corresponding invari- ant is / above. The second set contains the group generated by S, and the corresponding invariant is the product of * and tjf above ' The third set contains the group generated by UT2, and we get the invariant X = xi^^ - SSxi^xz"^ - ZSXi^Xz^ -{-X2^. These invariants satisfy the relation (E) Icosahedral Group. We shall take the group as repre- sented in Exercise 2, 103. There are three sets of subgroups of orders 20, 30 and 50, containing the groups generated by /', S'T' and S' respectively. We get, correspondingly, the three invariants T=Xi^^-\-X2^^+h22{xi^H2^-Xi^X2^^) - 10005(:i20a:2^o+xii0x220), E= -Xi^^-X2^^ + 22%{xi^H2^-XiH2^^)-AQ^Xi^^X2^^, J=X\X2{x\^^-\-\\Xi^X2^ X2^^), which satisfy the relation r2+^-1728/s = 0. EXERCISES 1. The invariant * of the tetrahedral group is the Hessian covariant (cf. 174) of the function *: 48V33*= *" *" and the invariant / is the Jacobian of the functions * and *; 4>i 4>2 -32V^/= ^2 Obtain similar relations for the octahedral and icosahedral groups. 2, From the fact that no two abelian subgroups of G can have a trans- formation in common (except similarity-transformations) unless they generate a larger abelian group, it follows that G is made up of a number of distinct abelian groups ^1, Ht, . . . , having no transformations in 105] THE FUNDAMENTAL INVARIANTS 227 common except E and Ei. Hence, if the orders of these groups be re- spectively Ai, . . . , we must have (1) g<t>=2*+(hi4>-2*)+(h24>-2)+ . . . Now, hi, . . . are factors of g, say hi=g/gi, . . . , and there are either gi/2 or gi subgroups conjugate to Hi, according as there is or is not a trans- formation in G which permutes the linear invariants of Hi. Hence, adding the corresponding terms in the right-hand member of (1), we obtain g0= 2 -I- ^ igi(A,-2) -f ^g'i(A',<^-2) or ^=\-x^HhxH} Verify this (Diophantine) equation for the groups (A) to (E). 3. Prove that, in the case of (D) or (E) , any invariant of degree g, say /, is an absolute invariant; that is, it is transformed into itself by every transformation of G. Prove also that / is a rational integral function of two of the three fundamental invariants listed above for the respective group. * Counting the transformations E and Ei once each. CHAPTER XI SOME SPECIAL TYPES OF GROUPS 106. Primitive and Imprimitive Groups. Let us suppose that the group G, 90, contains not only transformations of type A , but also some of type p q t V q p V t u w r s 5 = w u s r which upon the change of variables there employed becomes p-q t-v uw rs p-\-q t-\-v B = u-k-w r+5 fO B' B" then we say that G is imprimitive, under the assumption that it is transitive. In general, a transitive group G, in which the variables (either directly or after a suitable choice of new variables) can be separated into two or more sets Yi, . . . , Yt, such that the variables of each set are transformed into linear functions of the variables of the sawe set or into linear functions of the variables of a different set, is said to be imprimitive. If such a division is not possible, the group is primitive. The sets Fi, . . . , F* are called sets of imprimi- tivity, 228 107) IMPRIMITIVE GROUPS 229 107. Theorem 9. Let G be an imprimitive linear group in n variables. These may be chosen in such a manner that they break up into a. certain number of sets of imprimitivity Fi, . . . , Yt of m variables each {n = km), permuted according to a transi- tive substitution group K on k letters, isomorphic with G. That subgroup of G which corresponds to the subgroup of K leaving one letter unaltered, say Y\, is primitive as far as the m variables of the set Yi are concerned. If m = \, k = n, then G is said to have the monomial form or to be a monomial group. Proof. Let the variables of G break up into say k' sets Fi, . . . , Fjf, permuted among themselves according to a sub- stitution group K' on k' letters. This group K' is transitive (as a substitution group, 12); otherwise G would not be a transitive linear group. Hence i^'. contains k' l substitutions S2, S3, . . . , Sf which replace Fi by F2, F3, . . . , Yt' respectively. We shall select k' l corresponding transforma- tions of G and denote them by A2, A3, . . . , Af. The condi- tion that the determinants of these transformations do not vanish, implies that the sets contain the same number of variables n/k'. There is in K' a subgroup K\ whose substitutions leave Fi unaltered ( 12). This subgroup, together with the sub- stitutions S2, . . . , St' will generate K'. Correspondingly, G is generated hy A2, . . . , At> and that subgroup G\ of G corresponding to K' \, and which therefore replaces the variables of Y\ (say yi, >'2, . . . , >) by linear functions of the same variables. If we now fix our attention upon just that portion of each transformation of G which affects only these m variables and which plainly forms the transformations of a linear group [Gi] in m variables, we shall prove that if [d] is not primitive, then new variables may be introduced into G such that tlie number of new sets of imprimitivity is greater than k' . Accordingly, let the variables of [d] break up into at least two subsets of intransitivity or imprimitivity, say Fi"\ . . . , Y\:^ . New variables will now be introduced into the sets F2, . . . , Ffc* such that At will replace Fi"\ . . . , Y ^^ by dis- 230 SOME SPECIAL TYPES OF GROUPS [Ch. XI tinct subsets F/", . . . , F/*^. In this manner the variables of G will be divided into Ik' subsets, and it remains for us to prove that any transformation 5 of G will permute these sub- sets among themselves; that is, 5 will transform the variables from any one subset into linear functions of the variables of one of these subsets. Let 5 replace F^, by F^. Then AaSAfi~^ = T transforms Fi into itself; that is, T is a transformation of Gi and will therefore permute among themselves the subsets F/", . . . , Fi^". It follows that the transformation Aa~^TA0 = S will transform any subset of Ya into some subset of Yp, and the proposition is proved. We can therefore keep on changing the variables so as to increase the number of sets of imprimitivity, until the sets contain just one variable each, or until the group [Gi] is primi- tive. The theorem is therefore proved. 108. Lemma. A linear group G having an invariant abelian subgroup H whose transformations are not all similarity-trans- formations is either intransitive or imprimitive. Proof. Write H in canonical form. The variables can then be arranged into sets having the property that a transforma- tion of H affects all the variables of any one set by the same constant factor. To illustrate, let H be generated by the transformations Ti={fxi, ai, ai, ai, 02) (ais^aj), Ti=(pi, Pi, Pi, P2, Pi) (/3i7^/3i)' Here we have three sets: X= (xi, 0:2), Y= (xi, Xt), Z= (xt). Then it is readily proved that G permutes these sets among themselves. Thus, in the illustration given, let 5 be a trans- formation of G and r<, Tj of H, and let S-^TiS = Tj. Now suppose that the variables of X are transformed by S into two variables yi, y2 forming a set X'; we must then prove that X' is either X or F. We have (xi, X2)TiS = (xu X2)STj, or Oiiyi, y2)=(yi, y2)T,. 108] SYLOW GROUPS 231 Hence, the variables yi, y2 are transformed by Tj into the same multiple of themselves. Since Tf may be taken to repre- sent any transformation of H, it follows that yi, y2 are linear functions of xi, X2, or linear functions of X3, x^. Theorem 10. A linear group whose order is the power of a prime number can be written as a monomial group by a suitable choice of variables xi, . . . , Xn] that is, its transformations have the form: * Xs = astx't (s = l, 2, . . . , ; t = l, 2, . . . , n). Proof. 1. A group P whose order is the power of a prime number p is either abelian or it contains an invariant abelian subgroup Q whose transformations are not separately invariant in P ( 48). In the first case the theorem follows from Theorem 8, 96. In the second case, Q can be written in canonical form, and P is intransitive or imprimitive by the above lemma. If the theorem is true for a transitive group, it is evidently true for an intransitive group; hence we need merely discuss the case where P is imprimitive. 2. By Theorem 9, P is monomial unless there is a group [Pi] which is primitive in the m variables of a set Fi, But the order of [Pi] is again a power of p, and this group cannot there- fore be primitive, by 1. Corollary. A linear group in n variables whose order is the power of a prime greater than n is abelian. 109. Theorem 11. A linear group G in n variables and of order g = g'p^q^r^ > where p, q, r, . . . are diferent primes all greater than n-\-\, contains an abelian subgroup of order p"q^r<^ . . . To prove this theorem by the process of complete induction, we assume it true for any group whose order is divisible by a factor of pf^q^r^ , smaller than this number. We shall also assume the theorem true for a transitive group in fewer than n variables; it will then immediately be true for an intransitive group in n variables. * Proofs of this theorem were given by the author in Transactions of the American Mathematical Society, vol. 5 (1904), pp. 313-314; vol, 6 (1905), p. 232; and by Burnside, Theory f Groups, second edition, Cambridge, 1911, p. 352. 232 SOME SPECIAL TYPES OF GROUPS [Ch. XI We therefore assume that G is transitive. At the outset we anticipate a theorem given below (Cor. 3, 135), from which it follows that G contains a transformation of order pqr. . . . Now, among all abelian subgroups of G whose orders are of the form p^q^r^. . . . , where a^a^l, 6^/3^1, etc., let // be one whose order is the highest possible. We shall then prove that here (1) a = a, ^ = 1, 7=c, . . . For this purpose, let us assume that these equations are not all true; say a<a. Then H contains a Sylow subgroup P\ of order p" which is contained in a Sylow subgroup P of G of order p^ (11). The groups H and P being abelian (Cor., 108), all the transformations of P\ are invariant in both, and will therefore also be invariant in the group K generated by H and P. But, if such an invariant transformation is not a similarity-transformation, K will be intransitive (cf. proof of Theorem 8, 96) and will (by assumption) in this case con- tain an abelian subgroup of order p^q^r^ , contrarv to the supposition made in regard to H. Again, if the transformations belonging to P and invariant in K, say p' in number, are all similarity-transformations, the group K contains an invariant subgroup (Theorem 12, Ex. 2), the order of which is divisible by />""*. Hence, by assumption, it contains an abeb'an sub- group of order p''~'q^r'^ . . , The corresponding subgroup of K is evidently also abelian and has for its order p^-p'^~*(fr'^ . . . = p<^(ff . . . Hence finally, if a < a, i7 cannot be that abelian subgroup of highest order p"^r'' . . . contained in G. The equations (1) follow. EXERCISES 1. Prove that if n+1 is a prime, and if the order of G is g-g'in+lYp"^ . . . , where i> 1, and p, q, . . . are primes greater than n+1, then there is in G an abeh'an subgroup of order (n-t-l)VV 2. It follows from Theorem 11 that a group in n variables whose order contains no prime factors smaller than n-\-2 is abelian. Prove that if the order contains no prime factors smaller than + l, the group is abelian. 3. Construct all the types of (monomial) groups of order 2* in three variables, and show that such groups contain cither a transformation of order 8 or one of order 4 and type ( 1, i, j). 110] THEOREM ON SIMILARITY-TRANSFORMATIONS 233 4. Construct all the types of groups of order 3V(0= 1 or 3; cf. ( 111) and show that such groups contain a transformation of order 9<f>. 110. On the Group of Similarity-transformations. The necessity for the presence of similarity-transformations in a given linear group can be determined from the following Theorem 12. // a Sylow subgroup P of order p^ in a linear group G of order g can be generated by a group P' of order p^~^ and a similarity-transformation T of order p", then there is in G an invariant subgroup G' of order g/p which does not contain T. Proof. Let H represent the regular substitution group on g letters simply isomorphic with G ( 27). Then, if among the letters of H, yi, y2, . . . , )* form a transitive set for P', the sum J=yi-\-y2-\- . . . +yk will be transformed into itself by every substitution of P'. Let furthermore be a primitive root of the equation ^ 1 = (116); then the function /=/-i-rK/)r+r2(/)r2-f . . . -\-d-^+HJ)'P'-^ is transformed into a constant (d) times itself by T: iI)T=^{J)T-\-d-'{J)T^-\- . . . -\-d-'>+'{J)T^ = el+d[{J)v-j\ = dl, since 7^ belongs to P' and therefore transforms / into itself. Moreover, any substitution of P' transforms I into itself. For, p-i p-i p-i {i)s =^6-^)^8 =^e-V)ST=^d-V)'r, r=0 r = r-0 since T is commutative with S. It follows that / is an invari- ant of P (the function / cannot vanish identically, since no two of the terms J, {J)T, {J)T^, . . . can have a letter in common) . Again, / is not an invariant of any substitution in G other than those in P. For, if i? be a substitution such that {I)R-cI, the letters occurring in I must be permuted among themselves by R. Let us suppose that R changes yi into >'2. But, the 234 SOME SPECIAL TYPES OF GROUPS (Ch. XI letters of / form a transitive set for P; accordingly, there is a substitution in P, say ^i, which also changes y\ into y2. Then RSi~^ leaves yi fixed and must therefore be the identity; that is, R=Si. It follows that the substitutions of G transform I into just g/p*" functions I, Ii, I2, . > . , no one of which is a con- stant multiple of another. The product K = IIil2 ... of these functions is therefore an invariant of G. Now, all the substitutions of G for which iT is an absolute invariant, that is, for which {K)R = K, must form an invariant subgroup G' of G, as is easily seen. To this group the substi- tution T does not belong, since {K)T = ^K, . where k = g/p^. For, let If = {I)R, then {Ir)T = {I)RT = {I)TR = d{I)R = elr. But, ^9^1 since k is prime to p. This subgroup G' is of index p. For, the constant mul- tipliers of K that result by operating upon K by the various substitutions of G are integral functions of 6 and must be roots of unity. Such roots can therefore, by 116, 6, be no others than powers of 6. Moreover, it is readily seen that each power must occur equally often, so that the power 1 occurs g/p times. Hence the theorem. EXERCISES 1. Prove that a linear group in 3 variables of order 9g in which there is no transformation of order 94> must contain similarity-transformations. 2. Prove that a linear group in n variables which contains a subgroup P of similarity-transformations of order />' (/> a prime >), contains an invariant subgroup of index />*, to which F does not belong. CHAPTER Xn THE LINEAR GROUPS IN THREE VARIABLES 111. Introduction. As in the case of the binary groups we shall limit ourselves to the discussion of groups of trans- formations of determinant unity, and shall generally write g<l> for the order of a linear group whose corresponding collinea- tion group is of order g. Moreover, the order of a transforma- tion 5 will often be written in the form g4>, when the order of the group generated by S is g0. For instance, the order of the transformation S = (a, a, aw^) (a^ = w; w^ = l) may be written either 9 or 3<^. Though the orders as written may thus virtually refer to collineation groups, it must be kept in mind that all purely descriptive terms refer to linear groups. For instance, the group of order 90 generated by 5i = (l, W, W^), S2 = would be described as a non-abelian group, though the corre- sponding collineation-group is abelian. The determination of the linear groups in three variables is based upon the following classification: 1. Intransitive and imprimitive groups. 2. Primitive groups having invariant imprimitive sub- groups. 3. Primitive groups whose corresponding collineation groups are simple. 4. Primitive groups having invariant primitive subgroups. 235 1 1 ,1 236 THE LINEAR GROUPS IN THREE VARIABLES (Ch. Xil The method for the first two and last classes is more or less evident. In the discussion below on the groups in class 3 some theorems are developed that with slight modifications can be extended to groups in n variables, and such generali- zations are given in 126, together with a r6sume of results on the order of the primitive groups in n variables. An intro- duction to the theory of the invariants of the ternary linear groups is given in 125. 112. Intransitive and Imprimitive Groups. We have two t>'pes of intransitive groups: (A) X\=ax'\, a:2=/3a;'2, xz=^yx'z (abelian type). (B) xi=ax'\, X2 = ax'2+hx'-i, Xz=cx'2-\-dx'-i. In (B) the variables X2, x-s are transformed by a linear group in two variables (cf. Chapter X). The imprimitive groups are all monomial. There are two types: (C) A group generated by an abelian group H: xi=ax'i, ii[;2 = /3^'2, x^ = yx'z and a transformation which permutes the variables in the order (a;iX2a;3). By a suitable choice of variables this trans- formation can be thrown into the form Ti a;i=a;'2, X2 = x':i, xz=x'\. (D) A group generated by //, T of (C) and the transforma- tion R: xi=ax\, X2 = bx'3- X3'=cx'2- 113. Remarks on the Invariants of the Groups (C) and (D). Interpreting xi, X2, xs as homogeneous coordinates of the plane, the triangle whose sides are xi =0, a;2 = 0, r*;3 = is transformed into itself by the operators of (C) and (D); in other words. xia;2X3 is an invariant of these groups. It will later be imperative for us to know under what con- ditions there are other invariant triangles. Assuming the existence of one such, say (1) {aiXi-^a2X2-\-a3X3,){biXi-\-b2X2-\-b3X3){CiXi-{-C2X2+C3X3) =0, 114-1151 PRIMITIVE COMPOSITE GROUPS 237 we operate successively by the transformations of H and by T. Examining the various possibilities we find that (1) could not be distinct from x\X2Xii = unless H is the particular group generated by the transformations 5i = (l, CO, op), 52 = (a), w, w) (0,3 = 1). There are then four invariant triangles for (C), namely; . - XiX2X2 = 0; {Xi -\-X2-\- BXz) {X\ + (jiX2 + (JpOXz) {Xi + (jPx2 + OiBXz) = (0 = 1, CO or co^). The same triangles are invariants of (D) if this is generated by (C) in the form just given and the following special form of i?: Xl = X'\, X2=x'3, X-i=x'2' 114. Groups Having Invariant Intransitive Subgroups. All such groups are intransitive or imprimitive. This follows from the fact that the type (B) has a single linear invariant x\, which is therefore also an invariant of a group containing (B) invariantly; * and the fact that a group containing (A) invari- antly cannot be primitive by the lemma, 108. 115. Primitive Groups Having Invariant Imprimitive Sub- groups. It was shown above that the types (C) and (D) possess either one or a set of four invariant triangles. If they possess only one such triangle, a group containing one of these types invariantly would of necessity also leave invariant that triangle, as may be easily proved. That such a group may be primitive, it is therefore necessary that (C) and (D) possess the four invariants (2) . Let us therefore assume a group G permuting among them- selves the triangles (2), which we shall for brevity denote respectively hy h, t2, h, h'm. the order as they are listed in (2). * Let V be any transformations of a group containing (B) invariantly, and T any transformation of (B). Then VTV-^=Ti belongs to (B), and if we put {xi)Ti=cau {xi)V=y, we have iy)T={y)V-'T,V=ay, so that y is an invariant of Ti But, Xi being the only linear invariant for (B), it follows that y= cxi. 238 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII We now associate with each transformation of G a substitu- tion on the letters ii, tz, ts, U, indicating the manner th which the transformation permutes the corresponding triangles. We thus obtain a substitution group K on four letters to which G is multiply isomorphic, and the invariant subgroup (C) or (D) corresponds to identity of K. No one of the four letters could be left unchanged by every substitution of K; for, the corresponding triangle would be an invariant of G, and this group would not be primitive. Moreover, no transformation can interchange two of the triangles and leave the other two fixed, as may be verified directly. Under these conditions the following possible forms for K are found: (E') 1, {ht2){hk);* (FO 1, {ht2){hk), {hk){t2h), {hh){t2k)\ (GO the alternating group on four letters, generated by (/ife)(fe/4) and {t2kt^. Now, to construct the corresponding transformations we observe that the group (D) as given in 113 contains all the transformations which leave invariant each of the four tri- angles. Furthermore, we note that if a given transformation V permutes the triangles in a certain manner, then any trans- formation which permutes them in the same manner can be written in the form V = XV, X being a transformation of (D). For, F'F"^ must leave fixed each triangle, and is therefore a transformation X as defined. We are now in a position to construct the required groups. By direct application we verify that the transformations U, V, UVU-^: U: Xi=x\, :;2 = a;'2, X3=(>)x'3 {^ = (^); (3) V: xi = p(x\-\-x'2+x'3), X2 = p{x\-\-wx'2-\-<^x'3), X3 = p(x\-^<a^x'2-\-o}x'3) (p= -); \ (i) (a^/ UVU-^: xi = p{x'i-^x'2-\-(^x'3), X2 = p(x\-{-(ax'2+0}X'3), X3 = p{ux\ +X'2 + UX's) * The three different subgroups of order 2 oi K would furnish only one type for C, since the three different groups obtained are transformable one into the other by a t hangc of variables. 116] ON ROOTS OF UNITY 239 permute the triangles in the following manner: {t2hh) , (/1/2) (^3/4) , {hh) {t2h) . Accordingly, since all the required groups contain a trans- formation corresponding to {ht2){hU), every such group must contain a transformation XV, X belonging to (D). Hence, if G contains (D) as a subgroup, it also contains V. If, however, (C) were a subgroup of G, but not (D), then either V is con- tained in G, or else XV, where X belongs to (D) but not to (C). In this event X may be written XiR, where X\ belongs to (C). Hence finally, either V ov RV belongs to G. However, V^ = {RVy = R. Thus R, and therefore also V, are contained in G in any case. Again, if G contains a transformation corresponding to {t2ht^ or {tih{t2h), such a transformation can be written XU or XUVU~^, X belonging to (D). Hence, since G contains (D) as we have just seen, it will contain either U or UVU~^ in the cases considered. We therefore have the following types: (E) Group of order 36</> generated by (C) as given in 113: 5i = (l, CO, a>2), T: a;i=:;'2, X2=x'3, :r3=x'i, and the transformation V of (3). (F) Group of order 72<l> generated by Si, T, V and UVIJ-K (G) Group of order 216^ generated by Si, T, V and U. These groups are all primitive, and they all contain (D) as an invariant subgroup. The group (G) is called the Hessian group (cf . Jordan, Journal fiir die reine und angewandte Mathe- matik, Bd. 84 (1878), p. 209). 116. On Roots of Unity. A solution of the equation ^ = 1, n being a positive integer, is called a root of unity. A solution a 's in particular called a primitive th root of unity, if n is the least integer for which a'* = l. In such a case n is called the index of the root. Theorems. 1. The product or ratio of two roots of unity is again a root of unity. 240 THE LINEAR GROUPS IN THREE VARIABLES (Ch. XII 2. Any positive or negative rational power of a root of unity is again a root of unity. 3. If n is the index of a root a, and m a positive integer, the index of a" is n/d, where d is the highest common factor of n and m. 4. If the index of a root 6 is n = ah, where a and h are two integers which are prime to each other, then it is possible to find a root of index a, say a, and one of index h, say /3, such that As is customary, we write w, w^ for the roots of index 3; i, i for the roots of index 4 ; co, co^ for the roots of index 6, etc. 5. If a is a primitive wth root, then the n roots of a::" 1 = are a, o?, . . . , a"~S a", and we have l+a+a2+ . . . +a'-i=0. 6. Theorem of Kronecker. For the proper handling of a certain class of equations we use a very efTective theorem of Kronecker.* Instead of making a formal statement of the theorem we shall explain its meaning by implication. The class of equations referred to are all of the form S|lj[ai = 0; ai, . . . , at being roots of unity, and the ques- tion involved is this: if these roots are not known originally, but their number k is known, what can be inferred about their values? The theorem implies that the k roots fall into sets, each containing a prime number of roots the sum of which equals zero. Moreover, if p be the number of roots in any one of the sets, and if a be a root of index p, then the roots of the set are e, ea, . . . , ea""^, where e is an unknown root of unity. We shall discuss in full the cases ^ = 3, 4, 5. . ^ = 3: ai4-a2+3 = 0. Here we have a2=aiw, a3=aico^. ^=4: ai+a2+a3+a4 = 0. We have two sets of two roots each, say ai +a2 = 0, as -|-a4 = 0. k = 5: ai-|-a2+a3+a4+a6 = 0. There are two possibilities: one set only, or two sets containing 3 and 2 roots respectively. * Mfimoircs sur les facteurs iiT6ductiblcs de I'expression .t** 1, Journal de Alalliemaliqiics pares et appliqu6cs, scr. 1, t. 19 (1854), p. 178. 117] THE PRIMITIVE SIMPLE GROUPS 241 If /3 represents a primitive 5th root, and 7, 5 roots of unknown indices, the two cases are respectively given by 7+7/3+7/3^+7/33+7/3^ = 0; (7-7) + (5+5w+6co2)=0. By means of Kronecker's theorem the following can be proved : 7. If N represents the sum of a finite number of roots of unity and k an integer, and if it be known that N'/k is an algebraic integer (that is, a solution of an equation xf^-{-aixf^~^-\- . . . -\-am = 0, where ai, . . . , a^ are positive or negative integers or zero), then N/k equals the sum of a finite number of roots of unity. More definitely, the roots in N can be arranged into two sets such that the sum of those in one set vanishes and those in the other set are each repeated k (or a multiple of k) times. Primitive Groups whose CoRRESPoisroiNG Collineation Groups are Simple,* 117-123 117. Theorem 13. No prime p>7 can divide the order oj a primitive linear group G in three variables. Proof. The process consists in showing that, if the order g contains a prime factor p>l, then G is not primitive. We subdivide this process into four parts as follows: 1 proving the existence of an equation F = 0, where F is a certain sum of roots of unity; 2 giving a method for transforming such an equation into a congruence (mod p); 3 applying this method to the equation F = 0; 4 deriving an abelian self-conjugate subgroup P of order />*. 1. The order g being divisible by p, G contains a Sylow subgroup of order p^ and therefore a transformation 5 of order p. We choose such variables that 5 has the canonical form S = {ai, a2, as); ai'=a2'=3*= 1, aia2a3 = l. Two cases arise: two of the multipliers are equal, say ai=a2, or they are all distinct. They cannot all be equal, since ai'= 1 * We shall briefly call siu h groui^ primitive simple groups. 242 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII and ai^= 1 imply ai = 1, whereas S is not the identity. Of the two cases we shall treat the latter only: the method would be the same in the former case,* and the result as stated in Theorem 13 would be the same. Selecting now from G any transformation V of order />: bi V = a\ Cl 02 1)2 C2 as h cs . we form the products VS, VS^, VS*". Their characteristics ( 89) and that of V will be denoted by [VS], we have [V] =ai +62 +C3, [VS] =aiai -\-b2a2 -\-C3a3, [VS^]=aiai^+b2a2^-\-C3a3^, [75"] =aiai''+&2a2''+C3a3''. [V], and (4) (5) = 0. We now eliminate ai, &2, cs from these equations, obtaining [V] 111 [VS] ai a2 OL2, [F52] ai2 a2^ az^ [75"] ai" a2'' aa* Expansion and division by (ai a2)(a2 a3)(a3 ai) gives us (6) [F5'']+i<:[F]+Z,[F5]+iW[F52] = 0, K, L, M being certain polynomials in ai, a2, a3, with the general term of the type a.^a'^a^. Since ai, a2, as are powers of a primitive />th root of unity a ( 116, 5), the quantities K, . . . are certain sums of powers of a. Moreover, the characteristics [F] . . . are each the sum of three roots of unity (81, Ex. 7). If, therefore, the products in (6) were multiplied out, there would result an equation of the kind discussed in 116, 6. The * The congruence (10) would here be of the first degree in u- 117J THE PRIMITIVE SIMPLE GROUPS 243 various terms could therefore be rearranged in sets as explained in that paragraph, which gives us an equation of the form (7) ^(l+a+a2+ . . . +p-i)-f.5(i+^_|.^2 , , ^ 4.^-1) +C(H-7+72+ . . . +7'-')+ ... =0, A, B, C, . . . being certain sums of roots of unity; a, /3, 7, . . . primitive roots of the equations x^ = \, a:* = l, c^=\, . . . respectively; and p, q, r, . . . different prime numbers. The coefficients A, B, C, . . . may be put into certain standard forms. Thus, any root cj^l occurring in any of these sums will be assumed to be resolved into factors of prime- power indices (116, 4): = pet(r . , the root ep being of index p*^, e, of index ^, etc. Furthermore, within A any root p will be assumed to be either unity or a root whose index is divisible by p^. For, if it were of index p, say p=a*, we could put 1 in its place, since a*(l+a+a2+ . . . +0^-^) =*+*+' +a*+2_^ . . . +a*+'- = l+a+a2_|_ _ ^ _|_^-l by means of the relation a^ = l. Likewise we assume that any root cj within B is either equal to unity or is a root whose index is divisible by q^; and so on. To illustrate, let i be a root of index 4 and t a root oT index 9 (namely '=w), and let p = 2. q=3. Then the standard form for the expression (8) ( /- 1)(1-1)+(t^u - w^+i){l+u-\-w') would be 2. We shall now make certain changes in the values of the roots in the equation (7). First we put for every root ,, cr, . . . whose index 19 divisible by the square of a prime other than p^ (as 7^ in the example above), leaving undis- turbed the roots whose indices are not divisible by such a square, as a, o^, . . . , /3, . . . The quantities A, B, . . . are thereby changed into certain sums A', B\ . . . The equation (7) is still true, the vanishing sums l+a+o^-\- . . . -fo^"*, etc., not having been affected. 244 THE LINEAR GROUPS IN THREE VARIABLES (Ch. Xll Next we put in place of q 2 of the roots /3, /3-, . . . , /S*"*, and 1 for the remaining root, thus changing 5'(l+/3+)32+ . . . +/3-) into J5'(l+0+0+ . . . +(-1)), so that this product still remains equal to zero. Similarly, we put in place of r2 of the roots 7, t^, . . . , y*, and 1 for the remaining root, and so on. Proceeding thus, we shall ultimately change (7) into an equation of the form ^"(l+a+a2+ +a'-0=0, where A" contains roots of the form p only. Finally, we put 1 in the place of every root a, oc^, . . . , c^~^, as well as every root tp. The left-hand member may then no longer vanish, but will in any event become a multiple of p. The final value of the expression (8) would be (w + l)(l+l) = 2 or 0, according as is replaced by or L Notation 1. Any expression N which is a sum of roots of unity, changed in the manner described above, shall be denoted by N',. 3. We shall now study the effect of these changes upon the left-hand member of (6). Each of the characteristics [VS], . . . , [VS'^], being the sum of three (unknown) roots of unity, will finally become one of the seven numbers 0, 1, 2, 3, whereas [V], being the sum of three roots of index 1 or /> (cf. 1), will become 3. The left-hand member of (6) will thus take the form (9) [VS'^yp-\-3K'p+L'AVSY+M'AVS^Y and this number is a multiple of p (by 2). The values K'p, L'p, M'p may be obtained by treating them as indeterminates 0/0. Thus, K = 1 (ai a2)(a2 a3)(a3 ofi) ai 2 as ai^ ai^ a-^ ai** 02** a.s** We find i^'p=-KM-l)(M-2), r, = M(M-2), 3/',= -lp(M-l), 117] THE PRIMITIVE SIMPLE GROUPS 245 and if we substitute in (9) and multiply by /> 1 we obtain the congruence: (10) [VS''Yp=sn^-\-tn-\-v (mod/), s, t, V being certain integers, the same for all values of ft. We finally substitute in succession n = 0, 1, 2, . . . , p \ in the right-hand member of (10). The remainders (mod p) should all lie between 3 and +3 inclusive, the interval of the values of [F5'']'p. Now, each of these seven rem.ainders can correspond to at most two different values of n less than p, if s and / are not both =0 (mod p), by the theory of such congru- ences. Hence, there will correspond to the seven remainders at most 14 different values of n, so that p is not greater than 14 unless s=t = 0. Tr>'ing p = 13 and P = ll, choosing for s, t, v the different possible sets of numbers </> (the problem can be simplified by special devices)* we find that in no case can the remainders all be contained in the set 0, 1, 2, 3, unless 5 = /^0. Choosing therefore this alternative we get, if P>7, [VS'^Yp^v (modp). . In particular, [VS]'p^vMVYp = Z (modp), from which it follows that [F5]'p = 3. Again, from this equation we deduce that the roots of [VS] are of index 1 or p. For, if the index of one of these roots were divisible by the square of a prime, or by a prime different from p, then the changes indicated in 2 could be made at the outset in such a way that or 1 would take the place of this root. But then [F5]'p. would be one of the numbers 0, 1, 2, 3. 4. Accordingly, the product VS of any two transforma- tions both of order ^ is a transformation of order ^ or 1. The totality of such transformations in G, together with E, will * Since an-\-b runs through the p values 0, 1, 2, ... , ^1 (mod p) when fi does, we may substitute this expression for n in the right-hand member of (10) and select constants a,b so that this member takes a simpler form. For instance, if P=ll, the right-hand member of (10) may be reduce<l by this substitution to one of the forms M*+c; m; or c; according as s^O; ssO, tj^O; smImO (mod p). When p= 13 we get the forms m*+c; 2mHc; m; c. 246 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII therefore form a group P. The order of this group must be a power of />, since it contains no transformations whose order differs from /> or 1. Moreover, P is invariant in G, since an operator of order p is transformed into one of order p. Hence, G has an invariant subgroup P of order p*. But this subgroup is abelian ( 108, Corollary) and therefore G is intransitive or imprimitive ( 108, Lemma). Notation 2. A quantity iV, which is the sum of a certain number of roots of unity, in which every root e,, is replaced by 1, but in which none of the other changes indicated in 2 are carried out, will be denoted by Nj,. If iV = 0, then Np = (mod p). 118. Theorem 14. If a group G contains a transformation S of order p^<t), P being a prime >2, then there is an invariant subgroup Ep in G (not excluding the possibility G=Hp) which contains S^. Any transformation in H,, say T, has the property expressed by the following congruence: (11) [V]MVT], (modp), V being any transformation of G. In the case p = 2 the group G contains an invariant subgroup Hp if the order of S is p^, and S^* will belong to Hp,- also if S = { l,i,i),in which case S^ will belong to Hp. The proof follows the plan of that of the previous theorem. If p> 2, we write S in canonical form, and construct the products VS, VS^, VR, where R denotes 5*. Assuming that the three multipliers of 5 are all distinct, we obtain an equation cor- responding to (6) in 1: [VR]-\-K[V]-\-L[VS]-\-M[VS^]==0. However, the changes indicated in 2 are not carried out except that 1 is put for every root e, whose index is a power of p (cf. Notation 2 above). The coefficients L, M become multiples of p by this change, and we find K= \ (mod p). Hence finally, [VR]p-[V]p^O (modp). Now consider all the conjugates Ri, . . . , 7?^ to i? within G. They generate an invariant subgroup H, ( 14, Ex. 6), and 1191 THE PRIMITIVE SIMPLE GROUPS 247 they all have the property of T as expressed by (11), since they fulfil the conditions of the theorem. Moreover, any trans- formation T in //j, satisfies the congruence (11), since such a transformation can be written as a product of powers of Rif . . . y Rh. For instance, let T = RiR2, and we have [VRi]MV]v, [(VRi)R2]MiVRi)]p {modp). Hence, [VTl = [VR^R2]MVRl]MVV 119. The Invariant Group Up. The order of this group is a power of p. For, if its order contained a prime factor q, qj^p, there would be a transformation of order q in Hj,, say T. Then, if E represents the identical transformation, we have by (11), [Vl = [EV]ME]v=^ (modp). Hence, the multipliers of T being a, 0, y, we have a^+/3^+y=3 (mod p), and therefore (12) 2"V+^'+V)^32)--^. Unless T is a similarity-transformation (which we may assume it is not), the roots a, /3, y are not all equal. Hence, the left- hand sum is 2q or q according as a is or is not equal to one of the roots /3, y. The right-hand sum is, however, Sq or zero, accord- ing as a is or is not unity. It follows that the congruence is impossible except when p = 2, and a is one of the roots /3, y, or unity. Substituting ^~^ and y~' for a"-' in (12) we get similar results. Collecting these, we finally discover that in no case can q^^p. The order of Hp is accordingly a power of p, and the group is monomial (Theorem 10). The possibility G = Hp is accord- ingly untenable if G is primitive. Corollary. No primitive simple group can contain a Irans- formation of order p^<t> if p>2\ or p^ if p = 2. 248 TIIK I IMAR GROUPS IN THREE VARIABLES [Ch. XII 120. Theorem 15. No primitive simple group can contain a transformation S of prime order p, p>3, which has at most two distinct multipliers. Let S = {a\, ai, a2) * and assume first that /> = 7. This transformation leaves invariant a point (a:i=X2 = 0) and every straight line through it. This will also be the case with any other transformation S' conjugate to S (Theorem 3). There- fore, the line joining the two invariant points is invariant for both S and S'. If now the variables be changed so that the common invariant line is yi = 0, the group generated by 5 and 5' will be reducible and therefore intransitive, breaking up into a group in one variable (3'i) and one in two variables (>'2, >'3). But, there being no primitive or imprimitive finite linear groups in two variables generated by two transforma- tions of order p = l (cf. Chapter X), it follows that S and S' are commutative. Accordingly, all the conjugates to S are mutually com- mutative and generate an abelian group, which must be invari- ant in G (14, Ex. 6), and the latter cannot be primitive (108, Lemma). Next, let /> = 5. If 5 and 5' are not commutative, they gen- erate the icosahedral group (E), 103, in the variables y2, ya. This contains a transformation of order 3 whose multipliers are w, (j?, and a similarity-transformation whose multipliers are 1, 1 (cf. 97). The product of these two transforma- tions, as a transformation in the variables yi, y2, ya, can be written in the canonical form T = {\, w, oP). But such a transformation is excluded by the next theorem [put Si = r2 = (l, 0)2, co) and52 = r3 = (l, -1, -1)]. 121. Theorem 16. No primitive simple group can contain a transformation S of order pq, where p and q are different prime numbers, and Si = S^ has three distinct multipliers, while 52 = 5" has at least two. Let 5 be written in canonical form and assume 5i = (ai, a2, as). We then construct the transformations V, VS2, VSi, VS\^ and proceed as in 117, 1, obtaining an equation corre- * Such a transformation is called a homology. 122] THE rUIMITIVE SIMPLE GROUPS 249 spending to (5). After putting unity for every root of index />*, the equation becomes the congruence: \[VS2]p-[V]j>\{oci-a2){a2-a3){a,i-ai)=0 (mod p), which can be changed into the following: {[vs2l-[vuf^o after multiplying by a suitable factor, since (l-6)(l-62) . . . (l-e'-')=lim.^^=9 z-l Xl when is a primitive qth root of unity. Hence finally, [VS2]MVl (modp), and the argument of 118 is now valid. Corollary. No primitive simple group can contain a transformation of order 35, 150, or 21 <^. (If Si, representing respectively 5^, S^*, or 5^*, has not three distinct multipliers, Theorem 15 applies.) 122. The Sylow - Subgroups. Consider now a primitive group G of order g<t). A possible subgroup of order 5^ or 7^ would be abelian ( 108, Cor.). By trial we find readily that no such group can be constructed without violating Theorem 15 or the Corollary to Theorem 14. Again, if g is divisible by 35, we have a transformation of this order ( 135, Cor. 3). But this is impossible in a primitive simple group ( 121, Cor.), A subgroup of order 3*<^ is monomial, and contains an abelian subgroup of order 3*~*0 at least. Assume ^ = 3; we then have an abelian subgroup P' of order S^<f>. When we construct such a group, avoiding the invariant subgroup Hj, resulting from Theorem 14, we discover that it must contain a trans- formation of type T = (i, , (oi^), where ^ = w. Then if g is divisible by 5 or 7 at the same time, we would have a trans- formation of order 15</) or 21<^ ( 135, Cor. 3), violating the Corollary, 121. In any event, we can have no abelian sub- group of order 3*"^ if *>3 (cf. E.x. 4, 109). Finally, a subgroup of order 2* is monomial and contains an abelian subgroup of order 2*"* at least. By trial we find thatiSj-U2(cf. Ex.3, 109). 250 THE LINEAR GROUPS IN THREE VARIABLES (Ch. XII Collecting our results, we find that g is a factor of one of the numbers 23-33, 23-32.5, 23-32.7, and the question arises what simple groups can have such orders. Now, all simple groups whose orders do not exceed the largest of these numbers have been listed. There are four possi- bilities: g = 60, 360, 168, 504. 123. The Three Types of Primitive Simple Groups. The first two of the numbers just given are the orders of the alter- nating groups on 5 and 6 letters, and the last two the orders of certain transitive substitution groups on 7 and 8 letters (20). The last case may be excluded from consideration for the reason that we should here have a Sylow subgroup of order 8 which is abelian and generated by three operators each of order 2 (20, Ex. 3). But no such group can be con- structed in three variables, as may be found by trial. In the other c^ses the corresponding types may be con- structed after a set of generators and their generational rela- tions are given (cf. 103, and Ex. 2, ibid.). The variables are selected so that the generators appear in as simple a form as possible; for instance, some one of them is always written in canonical form at the outset. We shall omit the details here; it would afiford excellent practice for the student to carry out this work. (H) Group of order 60 generated by 1, 2, 3, satisfying the relations:* i3=22=32 = l, (i2)3 = (2^3)3 = 1, (i3)2 = l; namely: 1: X\=X2, X2=X3, ^3=i; 2 = (1, -1, -1); 3: Xi=^(-x\-\-H2x'2-\-Hlx'3), X2 = Ut^2x\-\-Hix'2-x'3), X3=^inix'i -x'2-{-fJi2x'3) ; * E. H. Moore, Proceedings of the London Mathematical Society, vol. 28, p. 357. 124] INVARIANT PRIMITIVE SUBGROUPS 251 where (I) Group of order 360<^, generated by i, 2, 3 of (H) and 4, where 42 = 1, {ExE^Y = {E2E,Y = {E'iE^f = 1 ; 4: :ri = ic'i, X2=0)X'3, X3= a)V2. (J) Group of order 168 generated by 5, T, R with the relations (cf. 20, Ex. 2) 57 = ^3 = 2^2^1, T-^ST=S\ R-^TR = 'n, {RSy = l; namely: S = (^, /32, /34); T: Xx=x''2., X2=x'z, X2,=X'\\ R: xi=h(ax\-\-bx'2-\-cx'3), X2 = h{bx\-{-cx'2-^(ix'3)) Xz = h {ex' i+ax'2+hx'z); where 1 /r=-^(^+^2_^^_^6_^5_^3) = V37- 124. Primitive Groups Having Invariant Primitive Subgroups. We have already determined the primitive groups containing an invariant subgroup of order 2^ or 3^ (114-115). The possibility of a subgroup Hj, shall therefore be excluded from consideration (cf. 119). Let us assume that the group (H) is contained invariantly in a larger group (H') of order GO/f</). In (H) we have 6 sub- groups of order 5, which must be permuted among themselves by (H'). One of them must therefore be invariant in a sub- group of (H') of order 6O/?0/6 = 10//(^. But no such group can be constructed if h> 1 without introducing the group lip. In this manner the cases (I) and (J) may be disposed of. There are left the cases (E), (F), (G). Now, these groups all permute among themselves a single set of four triangles. This set is therefore also permuted among themselves by a larger group including either (E), (F) or (G) invariantly. But the groups having this property were found in 115 to be just these three. 252 THE LINEAR GROUPS IN THREE VARIABLES [Ch. XII EXERCISES 1. Determine the linear groups in two variables by the method of this chapter. (To determine the primitive simple groups, show first that the order of such a group is a factor of 60<^). 2. Prove by the method of 120 that a primitive group in four vari- ables cannot contain a transformation whose order is greater than 5, if its characteristic equation has only two distinct roots. 3. Why is it necessary to add the statement " while 52=5* has at least two " (distinct multipliers) at the end of Theorem 16? 4. Obtain the group (H) by the method of Ex. 2, 103, in the following form: S={abcdc) = (l,e\ c); U=(ad){bc): a;i= 3c'i, a!;2= x's, X3=x'i', T=(ab)icd): Xi=-^ix\+x'2+x'z), V5 2 = -7^(2:>;'i +sx'2 +tx'z) , Xz = p(2x\ -fte'j +sx',) ; V5 V 5 where e5=l, S=e^+e\ t = +e*, Vl = t-S. Show also that (E) as given in Ex. 2, 103, transforms the variables yo= XiXi, yi = X2'^, T2= Xi^ into linear functions of themselves and hence appears as a linear group in three variables, which is precisely the group (H) as just exhibited if wc write yo, yi, yz for Xi, x^, Xz respectively in (H). 5. Obtain the group (I) by adding a transformation W={ad)(ef) to the list S, U, T of (H), Ex. 4, and show that W: Xi = ^(x\+\ix'2+\ix',), X2=-j:,(2\2x'i-\-sx\-\-txf,), V5 v5 Xz=--.{2X,x'i+lx'2+sx't); v5 where Xi = i(-lV"l45), X2 = i(-lTV^^). Tn this and the previous example we need not verify that the transforma- tions obtained actually generate the respective groups as soon as they have been fully determined from a number of arbitrarily chosen relations, since just one type of each group (H) and (I) has been shown to exist by the generational relations given in 123. (In determining W, account has to be taken of the fact that (I) contains similarity-transformations; cf. the determination of (E), 103.) 125 INVARIANTS OF GROUPS IN THREE VARIABLES 253 126. Invariants of the Linear Groups in Three Variables. The theory of these invariants is by no means so simple as the theory of the invariants for the groups in two variables, and it is beyond the scope of this book to enter into a general dis- cussion, except as to indicate the principal results. In all cases we have a set of fundamental invariants such that all others are rational integral functions of these.* We shall here give such sets for the groups (G), (H), (I) and (J). That they are invariants may be verified directly by the student. (G) If we introduce the abbreviations XiX2X3 = <f>, Xi^-\-X2^-\-X3^ = rp, Xi^X2^ -\-X2^X's^-\-Xs^Xi^ = X, the fundamental set consists of the functions f C6 = ^-12x; Cq = {xi^ X2^) (x2^ - X3^) {xs^ Xi^) ; Ci2 = ^(^ +21603). /)l2 = ^(2703-^3). which satisfy the relation {432Cg^-Ce^+SCeCi2y = 4(172SDi2^-\-Ci2^). (H) Here we have four fundamental forms. One of them is A =Xi^+X2X3, if we take the group as given in 124, Ex. 4. The other three can be obtained in a manner similar to that which we employed in the case of the group (E), 105. We select a linear invariant /i for each of a set of subgroups of (H) of orders 10, 6, 4, respectively, and form the products of the 6, 10, 15 different linear expressions into which /i is trans- formed by (H). Selecting, for instance, the group generated by S and U, Ex. 4, 124, for the subgroup of order 10, we get fi=xi and a corresponding invariant Bl=Xl{Xl-\-X2+X3){Xi+i'^X2-\-(X3){xi-\-X2+f*X3) ' {xi -\-^X2+e^X3) {xi -^e^X2 -h^Xs) = Xi (0:2* +a;3* +5^:1X2^X3^ 5:ri%2:JC3 +iCi*) * Cf. Dickson, Algebraic Invariants (Wiley & Sons), New York, 1914, p. 70 ff. t Maschke, Mathemalische Annalen, Bd. 33 (1889), pp. 32.'>-.326. 254 THE LINEAR GROUPS IN THREE VARIABLES ICh. XII These invariants must evidently be transformed into the functions/, E^ T of the group (E), 105, by the substitution of X2X\, X2^j Xi^ for xi, X2, xz respectively (cf. Ex. 4, 124). The relation among the invariants A, B\, . . . corresponding to that connecting /, H, T is of degree 30 in the variables Xi, X2, X3, and its determination may be left as an exercise for the student.* (I) The fundamental invariants are four in number, f One is the function F=A^-\-\Bu where (taking for (I) the group given in Ex. 5, 124) ^4 and Bi are the functions listed above under (H), and X = 9dz3V-15 20 The three other invariants are obtained by evaluating the Hessian of F: Fn F\2 Fl3 x= F21 F22 F22. f F31 F32 F33 then the bordered Hessian : Fn F12 Fi3 Xi F = F21 Fai F22 Fs2 F23 F33 X2 X3 X, X2 X3 and finally the Jacobian of F, X, and Y. (J) As in the previous case, the fundamental invariants are four in number; J a function J=Xt?X3 -\-X2^xi +:*:3^iC2 ; the Hessian of / (say X) ; the Hessian of / bordered by the first derivatives of X (as Y above); finally, the Jacobian of the three functions already determined. * Cf. Klein, Vorlesungen Uher das Ikosaeder, Leipzig, 1884, p. 219. t Wiman, M alhemalische Annalen, Bd. 47 (1896), pp. 548-556. i Klein, M alhemalische Annalen, Bd. 14 (1879), pp. 446-448. 126) ORDER OF A PRIMITIVE GROUP 255 EXERCISES 1. Construct a fundamental set of invariants for the groups (E) and (F). 2. The invariant Dm under (G) is the product of the four functions U, k, ti. It, 113. Are there other relations among the invariants C, Ca, Ci2, Di2 and the functions /i, h, tz, i*? 126. Order of a Primitive Group in u Variables. In con- clusion we shall state some general results bearing on the order of a primitive group G in w variables. The Theorem 13 admits of generalization to n variables as follows. The number of the equations (4) would now be + l, the left-hand members being [V], . . . , [75""^] and [75"]. EHminating ai, 62, . . . , we obtain the corresponding equa- tion (5), from which a congruence similar to (10) is derived: [r5'']'p=5M''-'+/M'*"^+'t'M''"'+ . . . (mod p). In this we substitute in succession )u = 0, 1, . . . , p 1. The left-hand members are integers having the 2w-|-l values rang- ing from n to +w inclusive, and for each such value we can have at most 1 values of n by the theory of congruences, unless the right-hand member is merely a constant. Hence, excluding this possibility, there are at most ( l)(2w-f-l) values of n corresponding to the values assumed by the right- hand members; accordingly, if p> {n l){2n-\-i), the right- hand member is a constant merely, or [VS'^YMVYp^S (modp). From this it follows as in 117, 4, that the group has an invari- ant abelian subgroup of order />* and cannot be primitive. Hence, tite order of a primitive linear group in n variables can- not be divisible by a prime greater than (w 1)(2-|-1). The Theorem 14 may similarly be generalized to read: if a group G contains a transformation S of order ^4>^pn(t>, then there is an invariant subgroup lip in G whose transformations have the property [V]MVrl (modp), where V and T represent any transformations of G and H, respect- ively. )r.) and its order is ^5"^ emaining factor of g is the 50 ^5"-^ It follows that variables is oved the classical theorem that ablcs is of the form Xf, where / ibgroup, and where X is inferior upon n (Journal fiir die reine Definite limits to X have been laracteristics belong to a given oniglich Preussischen Akademie Jieberbach and Frobenius {Sitz- fT); and by the author {Trans- ety, vol. 5 (1904), pp. 320-321 5 for imprimitive groups). The CHAl THE THEORY OF G 127. Introduction and Del by Frobenius and is largely Burnside | simplified the th( tions. We shall devote thii main points of the theory 258 GROUP CHARACTERISTICS ICh. XIII A function /(xi, . . . , x) which is transformed into a con- stant multiple of itself by every transformation of G is called an invariafit of G. It is an absolute invariant if the constant multiplier is unity for every transformation of G; otherwise it is a relative invariant. A series of invariants /i, - - - , ft are said to be independent of each other if the variables xi, . . . , Xn cannot be eliminated from the equations fi=ai, . . . , /t = a*, where ai, . . . , ai are arbitrary constants. They are said to be linearly independent if no identity exists of the form 61/1+ . . . +Vt=o, where bi, . . . , bt are constants, not all zero. We shall as hitherto denote by (f)S the result of operating upon a function / by a transformation S. 128. Theorem 17. The number of linearly independent absolute invariants of G of the first degree in xi, . . . , Xnis it- t-i Proof. 1. If Sxt5^0, then G will have an absolute invariant of the first degree. For, let^'i, . . . , yn be arbitrary constants, and let/=yia;i+ . . . +ynXn. Then the function (/)5i+ . . . -\-{f)S,^F will be an absolute invariant of G provided that it does not vanish identically. This is seen as follows. We have (F)5, = (/)5i5,+ . . . +(/)5^r = (/)5i+ . . . +(f)S,=F. Thus, F is an absolute invariant unless it vanishes. If it does, then 2xi = 0. For, the sum of the coefficients of yixi, y2X2, ' , ynXn in {f)St IS readily found to be xt- Hence, at least one of these terms will be present inF if 2x5^0; say yixi. If therefore we put yi = l, ^2= . . =yn=0, the func- tion F will contain a term involving xi and hence it does not vanish. 128] NUMBER OF INVARIANTS 259 2. Let us now suppose that G has just k linearly inde- pendent absolute invariants of the first degree. Wc may assume that the n variables were chosen originally such that xi, X2, . . . , Xk are the invariants in question. Then G is intransitive, breaking up into ^ + 1 sets of intransitivity, containing respectively 1, 1, . . . , 1; n k, variables: Xi, X2, , Xk-, (Xk+i, . . . , Xn), by Theorem 6. Thus, if M = 2, ^=1, and if Xi is the absolute invariant, the matrix of any transformation of G will be of the form 1 o" c d The reducible group may now be transformed into an intransitive group of the following special form 'l d The characteristic xiSt) is accordingly equal to k-\-x(S't)y where S't is the transformation corresponding to St in that com- ponent of G which involves the set (a:t+i, . . . , Xn). Now, Ttx{S\) =0, or we would have a new invariant by 1. Hence, which completes the proof. Corollary 1. The number of linearly independent abso- lute invariants of degree m in xi, . . . , Xn is jX^(sn, where x(Si^^^) represents the sum of the homogeneous products of degree m in the multipliers of St, namely ar+a2'"+ . . . +ai'""^a2+ . -\-ar-^a2a3-\-. . . Proof. For brevity we take n = m = 2. When the variables Xi, X2 are subjected to a linear transformation b] S = 260 GROUP CHARACTERISTICS [Ch. XIII the products of x\, x-^ of the second degree, namely Xi^, xiX2< X2^, are correspondingly subjected to a linear transformation 5< = a2 2ab b'' ac ad-\-hc bd c2 2cd d^ and to a group G of transformations S, . . . will correspond an isomorphic group G^^ of transformations S^^\ .... If 5 is written in canonical form {a, /3), so is S^^\ namely (o^, ajS, /S^). Hence xC-S^^^) =^+a,5+/32. We now apply Theorem 17 to the group G^^^ and obtain the Corollary 1 for the case n = m = 2. Corollary 2. In the case of a transitive group, 11lZlxt = ^^- For, in this case there are no invariants of the first degree. By an elaboration of this principle we obtain the important result that if *< represents any given integral symmetric function with integral coefficients of the multipliers of St, then 2j<Ii*=/g, where / is a positive or negative integer or zero.* EXERCISES 1. Write a given substitution group as a linear group so that the letters of the former are the variables of the latter, and determine the characteristics. 2. Prove that the average number of letters which remain unchanged by a substitution of a transitive substitution group G is equal to unity. (Prove first that G contains a single absolute invariant of the first degree.) 129. Lemma. Let there be given a function /=ZiFi+ . . . +XtYt, where Xi, . . . , Xt are linear functions oi xi, . . . , Xn, the variables of a group G', and Fi, . , . ,Yt linear functions of yi, . . . , ym, the variables of a group G" simply isomorphic with G'. We may assume that Xi, . . . , Xt (as well as Fi , . . . , Ft) are linearly independent of each other; if it were possible to * See Transactions of the American Mathematical Society, vol. 5 (1904), p. .464ff. 130] TRANSITIVE GROUPS 261 write, say, Xk = aiXi-h . . . +at-iA'*_i, the function/ could be expressed in fewer than k terms: A'i(Fi+aiFt)+X2(F2+a2Ft)+ . . . + Xt-iiYt-i+at-iYtY = XiF'i+X2F'2+ . . . +Xt_,F'*-i. Now if f be unaltered when operated upon simultaiuously by corresponding transformations of G' and G" (in otlier words, if f is a bilinear invariant), and if k<n, G' is intransitive. Proof. For the sake of simplicity take ^ = 3. The variables of G' and G" may be so chosen tlndii f =x\yi-\-X2y2-\-xzy:i. A transformation of G' will change x\, X2, x^ into three linearly independent functions of a;i, . . . , Xn, and the corre- sponding transformations of G" will change yi, y^, >'3 into three hnearly independent functions of yi, , . . , y^. Let the resulting expression be /' = X'lF'i + X'2F'2 -f X'sF's, and we should have /=/' But this implies that X'\, X'2, X'3 are linear functions of xi, X2, X3. Hence, if w>3, G' is reducible and accordingly intransitive. 130. Theorem 18. Representing by x the conjugate-imag- inary of Xt we have, for a transitive group G, V UXt = g. Proof. Let G be th_e conjugate-imaginary group of G (92). Then, if 5 and S are corresponding transformations of G and G: 6 ... 1 (ab.. 5 = d . S = the n^ products xiX\, xiXj," . . ^ , a:^x are subjected to a corre- sponding linear transformation S' belonging to a group K isomorphic with G and G: _ ' aa ab . . S' = ac ad . . 262 GROUP CHARACTERISTICS [Ch. XIII The characteristic of S' is the product of the characteristics of 5 and S: x{S') = x{S)x{S) (this is seen readily when S (and therefore 5 and S') is written in canonical form). Hence, by Theorem 17, the number of linearly independent absolute invariants of the first degree in the variables of K is Tixixt/g. Any such invariant can be thrown into the form f=XlXl+X2X2+ . . . +X^, where Xi, . . . , X are linear functions of o;;!, . . . , Xn. We know one such invariant already, namely the Hermitian invariant ( 92), and we may assume the variables originally so chosen in G and G that this invariant is I=XiXl-\-X2X2-\- . . . +0C^. Then, if X be any constant, the expression f-h}J=(Xl + 'KXi)Xi-\-{X2-\-\X2)x2+ . . . +{Xn-^'^n)^ is also an invariant. Now, the constant X may always be determined such that Xi+Xrri, X2+Xa;2, . , X+X:r are not linearly independent. Therefore either G is intransitive by the lemma above, or /+X/ vanishes identically. Hence, since the first alternative violates the assumption of the theorem, any invariant / of K is merely a constant multiple of / (viz., /= X/); in other words, the number 2 x<Xt/g of linearly independent invariants / is unity. The theorem follows. EXERCISE Prove that if G is intransitive, ^xtxt=lg, where lis a. positive integer greater than 1. 131. Equivalence. Two simply isomorphic groups are equivalent if a suitable change of variables in one will make the matrices of their corresponding transformations identical. If no such choice of variables is possible, the groups are non- equivalent. For example, the groups generated by the transformations r 5= 1 T=il, -1) 131] EQUIVALENCE 263 are equivalent. If we put yi = Xi+xi, 3'j = xi jcj in S, then this transfor- mation takes the form (1, 1) in the variables yi, y,. Theorem 19. Let G' = {S'uS'2, . . . , 5',) andG" = (S"i, S"2, , S"g) be simply isomorphic linear groups in n and m variables respectively, and lei G' be transitive. Then if S"t repre- sents the conjugate-imaginary of the transformation S"t, X t-1 x{s\)x{s";)=kg, where k = or a positive integer. If k = l, G" is equivalent to G'; if k>l, G" is intransitive, and in this case k of its sets of intransi- tivity are transformed according to k groups each equivalent to G'. Proof. Let x\, . . . , Xn be the variables of G' and y\, . . . , 3'fl, those of G". We construct the group K in the nm variables Xiyi, . . . , xym, where yi, - - - , ym are the con- jugate-imaginaries of the variables of C. Applying Theorem 17 we find k linearly independent absolute invariants of K all of the form anxiyi-j- . . . -{-anmXnym- By a suitable change of variables in G" we now cause one of these invariants to become xiyi-\-X2y2-\- . . -\-Xnyn, and comparing this with the Hermitian invariant xi:ri+ . . . -i-XnXn of G' we may readily prove that G" transforms the variables yi, - . , yn among themselves according to a group equivalent to G\ Hence, if ^ = 1 and m = n, G' and G" are equivalent; if ^^ 1 and m>n, G" is reducible and therefore intransitive. Conversely, if G" is known to break up into sets of intrans- itivity, k of which are transformed according to groups which are equivalent to G', then the conjugate-imaginaries of the variables yi, . . . , y of any one of these sets will combine with xi, . . . , Xn to form one invariant Xiyi-\- . . . + Xiiy^ for K, making k such invariants in all. Corollary 1. If G' and G" are equivalent, their correspond- ing characteristics are equal, and x(5'i)-x(5^)+ . . . +x(5'a)-x(5^.)=g, if they are non-equivalent and are both transitive, this sum vanishes. 264 GROUP CHARACTERISTICS (Ch. Mil Corollary 2. Let G be a transitive linear group of order g in n variables, and let 17 be a regular substitution group ( 27) on g letters simply-isomor phic with G. Then if II be looked upon as a linear group in g variables, it is intransitive, breaking up into a number of component groups among which are found just n which are equivalent to G. Proof. When a substitution Tt of H other than the identity is written in matrix forrti as a linear transformation, every element in the principal diagonal is zero, since otherwise the corresponding letter would be replaced by itself in Tt. Accord-, ingly, x(3^t)=0 unless Tt is the identity; if Ty is the identity: (1, 1, . . . , 1), then x{Ti)=g. The transformations of G being correspondingly St and 5i = (l, 1, . . . , 1), we have therefore (-1 x{St)-x{Tt)=ng. The corollary now follows by applying Theorem 19. 132. Remark. The propositions of 131 become wider in scope by an obvious extension of the concept " group." A group G' of order g' to which another group G of order g = hg' is multiply isomorphic may be exhibited in such a way as if it were a group simply isomorphic with G, namely by repeating each of its transformations // times. For instance, the sub- stitution group of order 6: 1, {ab), (ac), (be), (abc), (acb) is multiply isomorphic with two of its subgroups: 1; and 1, (ab). With the concept of " group " extended as indicated above, we may exhibit the three groups as simply isomorphic in the following manner: (ac), (be), (abc), (acb); 1, 1, 1, 1; (ab), (ab), 1, 1. EXERCISES 1. Prove that if the regular substitution group H is broken up into its ultimate sets of intransitivity with their corresponding componeni groups, and ii H he multiply isomorphic with a transitive linear group 6 1, (ab), 1, 1, 1, (ab), 133] SUM OF MATRICES 265 in n variables, then n of the comiwnent groujjs into which // breaks up are equivalent to G. 2. The group // of order g is always multiply isomorphic with the group consisting of the identity alone, which is a transitive linear group in one \ ariable: x=x'. Hence, one of the sets of intransitivity of H will contain one variable, and the component group will consist of the identity repeated g times. Prove this in another way by showing that // possesses a single absolute invariant of the first degree. Prove also that if an additional set of intransitivity of H contains one variable, then H possesses a relative invariant ( 127) of the first degree. In such a case H is not a simple group; all those of its transformations for which this invariant is absolute form a self-conjugate subgroup. 3. Among all the component transitive linear groups into which H breaks up, let there be k which are non-equivalent: Gi, Gi, . . . , Gi in i, 2, . . . , Hjt variables respectively. Prove that g=i^+n2''+ . . . 133. The Sum of Matrices. The sum of a series of square matrices of the same order is the matrix whose elements are the algebraic sums of the corresponding elements of the given matrices. Thus, 'ai hi + '^2 h2 ^ Cl di. C2 d2. Ci-\-C2 di-\-d2 li Si, S2, . . are linear transformations in the same vari- ables, we shall write ^i +52+ ... to denote the matrix which is the sum of the matrices of 6*1, 52, . . . Multiplication of matrices is carried out according to the rule given in 78, irrespective of whether the matrices repre- sent hnear transformations or not. We find Si +52 =52 +5*1, nSi+So-h . . . ) = r5i+r52+ . . . , r-K5i+52+ . . . )T = T-'SiT-{-T-^S2T+ . . ., (5i+52+ . . . )(ri+r2+ . . . ) = 5iri+5ir2+52ri+52r2+ . . . . If M represents a matrix, and c a constant or a variable, the s>Tnbol cAf shall represent the matrix obtained by multi- plying every element of M by c. 266 GROUP CHARACTERISTICS (Ch. XIH 134. Lemma 1. If Si, . . . , Sm are the different trans- formations of a conjugate set of a transitive linear group G in n variables, then the matrix M=Si+ ... +5. is commutative with every transformation of G and has the form of a similarity-transformation {a, a, . . . ,a), where a = mx{S\)/n. Proof. That M is commutative with any given transforma- tion r of G is seen as follows. We have r-wr=r-i5ir-fr-i52r+ . . . =5i+52+ . . . =m, since T'^^iT, . . . , r'^^mr are the transformations .Si, . . . , Sm over again in some order. Hence MT = TM. Again, that M has the form of a similarity-transformation can be proved in several ways. We shall here give a very simple proof based upon the proposition, following from Theorem 21 : the w- elements of each of the matrices of the transformations of G do not satisfy a linear homogeneous equation whose coefficients are the same for every transformation. Now, the condition MT = TM is readily found to imply just such an equation, unless M is in the form of a similarity-transformation (a, a, ...,a). Finally, to find the value of a we observe from the formation of M that the sum of the elements of its principal diagonal, na, equals the sum of the characteristics of ^i, . . . , Sm- But these are all equal ( 89); hence a = wx('S'i). Lemma 2. If Mi, . . . , Mn are the matrices representing each the sum of the matrices of a conjugate set of G, there being h such sets, then MMt = Cu\Mi-\-c^2M2-\- . . . +CstnMn {s,t = \,2, . . .,h), where the coefficients Cai, . . . are positive integers or zero. Proof. Let M.=5i-f52 . . . -h5, M,=Ri-{-R2-h . . . +R then the gsgt matrices in the product MsMt = ljSxR^ must make up one or more conjugate sets, since T-'{M,M,)T = iT-'M,T){T--'MtT)=M,Mu 1351 PRODUCT OF CHARACTEKISTIC8 267 Accordingly, this product is the sum of one or more of the matrices M\, . . . , Mu, possibly repeated a certain number of times. Hence the lemma. 136. Theorem 20. Let the number of transformations in the different conjugate sets of a transitive group G in n variables be g\, g2, . . . , gh, and let the corresponding characteristics be denoted by xi, X2, . . , Xa (cf. 89). Then (') (f^)(^)=i:-fv) (^. '=>. 2. . *). where Ca\, . represent certain positive integers or zero. Proof. We substitute in the equation of Lemma 2 the canonical forms of the matrices Afi, . . . , Mu as given by Lemma 1, and obtain the equation (/3, . . . , /3) = (7, . . . 7), where /3 has for value the left-hand member of (1), and 7 the right-hand member. EXERCISES 1. Selecting the h equations (1) obtained by keeping 5 fixed while taking t=\,2, . . . , h, prove that gsxs/n is an algebraic integer (cf. 116, 7). 2. Prove that if Ss and Sr^ are conjugate, then k t i= 1 >= 1 ^' where the summation extends over a set of non-equivalent groups into which the regular substitution group H breaks up; if 5, and St~ ' are not conjugate, the first sum vanishes. (Prove that 2^*iXi>^ = if Sv is not the identity; and that if it is, the sum equals g. Prove also that if 5s and Sr ' are conjugate, gs=gt, and in the right-hand member of (1) we shall then have cai = gs- We assume nj=xi^ to be the characteristic of the identity.) Corollary 1. The quantity gx equals the product of n by the sum of a finite number of roots of unity. This follows from the statement of Exercise (1) and 116, 7. Corollary 2. The number of variables n of a transitive linear group G is a factor of the order g. Proof. The equation from Theorem 18 may be written (2) g =1:1X1X1+^2X2X2+ . . . +gX*XA. 268 CiKUUl' CHAKACTEKI8T1C8 [(Ji.Xiil Now, since the sums and products of algebraic integers are again algebraic integers, the quantity s.=jiixT+^^3ei+ . . . +^r n n n n is an algebraic integer. It follows that g/ti, being a rational number, must be an ordinary integer. EXERCISE 3. Prove that if a transitive linear group G of order g in n variables contains a subgroup of order / comf)osed of similarity-transformations, then g is divisible by/n (Schur). (Prove first that if x* does not vanish, there will be / distinct conju- gate sets for which the products gsX^Xs in (2) have the same value.) Corollary 3. If a transitive linear group G in n variables contains two characteristics Xs, Xi suck that the sum of the w^ roots in the product XsXt cannot be written as a sum in which primitive roots of index k are absent, theft there is in G a characteristic con- taining roots of index k and therefore a transformation whose order is k or a multiple of k* This follows from the equation (1). By the conditions of the corollary, at least one of the characteristics Xc of the right-hand member must contain roots of index k. There is, therefore, a transformation whose order is divisible by k. For, the order w of a transformation 5 = (a, /3, . . .) is the least common multiple of the indices of the roots , /3, . . . , since 5- = (l,l, . . . ) = (a,r, . . . ). To illustrate, let Xa= 1+i+i and Xta+a+c^, where t = V 1 and a is a primitive fifth root. Here xXt, or Aia-\-2ic? 2ao?, cannot be written as a sum which is free- from roots of index 20 (fa, etc.) by Kronecker's theorem ( IIC). EXERCISE 4. Prove that if a group in n variables contains transformations of orders p and q, two difTerent prime numbers both greater than + l, then the group contains a transformation of otder pq. 136. Theorem 21. Let G = {Si, 52, ... , Sg) be a transi- tive linear group in n variables. Then the n^ elements in tlie * Bumside, Theory of Croups, second edition, p, 347. 136) GROUP-MATRIX 269 matrix yiSi-i-y2S2-\- . . -^-yoSt ( 133), considered as func- tions of the g independent variables yi, . . , >*, are linearly independent. Proof. 1. When a square matrix M of n^ elements is transformed into a similar matrix M' by means of a linear transformation T in n variables: T-'MT = M', M = TM'T-\ then the elements of M' are linear functions of the elements of M, and vice versa; the coefficients being functions of the elements of T. Hence, if the elements of Af are linear functions of certain independent variables, the elements of Af' will like- wise be linear functions of these variables; and, if among the former just / are found to be linearly independent, the same will be the case with the elements of M'. 2. Now consider a regular substitution group H of order g written in the form of a linear group. As an example we take the symmetric group on 3 letters, which may be written as a regular group on 6 letters xi, . . . , xc as follows: Si = the identity, S2 = (:*;iX2iC3) (:r4a;5^6) , 53 = (iCiX3af2) (xiXeXs), S5 = (iCiXs) (0:2X4) {X3X6) , The matrix yi5i+y252+ . 54 = ixiX4,) fe^e) (iCaXs), 56 = (xiXe) (a:2:r5) ix3X4). . +3'65'6 is here M = yi y2 y-s >'4 ys ye ys yi y2 ys ya 3'4 y2 ys yi ye 3'4 >'5 3'4 ys ye yi y'2 ya ys- ya y4 y3 yi y2 , yo >'4 ys y2 y-s y\ New variables may be introduced in ^ so that this group takes the intransitive form, namely Zl=Xi-\-X2+X3-\-X4-\-X5-\-X6, Z2=X\+X2-\-X3X4Xs Xli, Zz=Xi-\-uX2-h 0x^X3, Zi=X4-\-orX5 + <j}Xa, Z5=X4-\-uX5-\-ur^XQ, ZQ=Xi-\-urX2-\-OiX3, 270 GROUP CHARACTERISTICS (Ch. XIII B c D E F C D E F where w is a primitive cube root of unity. When H is written in the new form (we shall denote this by H', and its transforma- tions by 5'i, . . . , S'q), the matrix M will become A 0' M' = where A=y\+y2-\-yi-\-y^-\-y5+yQ, B=yi-{-y2-\-yi-y4.-y5-yQ, C=3'i+w2y2+w>'3, D=y^ + o)y5-\-(^yQ, E = y^-\-o)'^y5+(ayQ, F = yi-\-oiy2-\-o)^y^. Now, there is a transformation, T say, which transforms H into fl^' ( 88) : T~^SiT=S'\, Qtc. Therefore, since r-(yi5'i+ . . . +yoSg)T=yiS'i+ . . . +y^' we have T~^MT = M'. Hence, by 1, there are as many linearly independent functions of yi, . . . , y^ among the ele- ments A, B, C, D, E, F of M' as among the elements of M, namely 6. It follows that A, . . . , F are all linearly inde- pendent. In the general case, H' breaks up into k non-equivalent groups Gi, . . . , Gt, with G, repeated as an equivalent group fit times, namely the number of its variables ( 131, Cor. 2). Correspondingly, the , component matrices involved in M' will have the same elements. Hence, there will be at most as many independent elements in M' as are found in a set of non-equivalent groups, namely ni^-\-n2^-\- . . . -\-ng^. But this number is equal to g ( 132, Ex. 3), the number of inde- pendent elements (yi, . . . , yo) in M*. Hence, by 1, all * Each element of M will contain a single variable yj. This is proved easily from the facts that ^ is a transitive substitution group and every one of its sub- stitutions (except the identity) displaces all of the g letters. 1371 NUMBER OF NON-EQUIVALENT GROUPS 271 these nr^-\- . . . -\-nt^ elements are linearly independent, and the theorem is proved. EXERCISES 1. Prove that the n' elements of each of the matrices of the trans- formations of G do not satisfy a linear homogeneous equation whose coefficients are the same for every transformation (Burnside). 2. Prove that if a certain element Cst vanishes in every transformation ^= [cafi] of a group G, the subscripts s, t being given, then G is not transitive (Maschke). 137. Theorem 22. The number (k) of non-equivalent transitive linear groups into which the regular substitution group H breaks up (cj. 131) is equal to the total number of sets of conjugate substitutions of H. The proof follows that of Theorem 21 closely, after we have first made equal to each other those of the variables y\, . . . ,yg which are factors of conjugate transformations in the matrix M. If, therefore, G contains h conjugate sets of respectively gi, . . ' , gk transformations, we shall have // independent variables, say vi, . . . , Vt,. The matrix M' now has the form of a transformation in canonical form. Thus, the matrix M' in the example given in 136, 2, becomes M' = {A, B, C, C, C, C), where A =Vi-\-2V2 + SV3, B=Vi-\-2V2 SV3, C = ViV2. In general, let Gj be any one of the groups into which H breaks up, and xi, . . . , Xa the characteristics of the various con- jugate sets of Gj. Then it follows by Lemma 1, 134, that, as far as the variables of Gj are concerned, M' will appear in the form of a similarity-transformation (/3/, /3/, . . . , /3;), where 0i = (giViXi-\-g2V2X2-h . . . +ghVhXH)/n. If Gj and G'j are equivalent groups, /3/ = /3'^; and conversely ( 131). Hence, if Gj and G'j are non-equivalent, fijj^fi'j. Accordingly, among the g multipliers of M' in its new form, there will be just k that are distinct, and these can certainly not furnish more than k expressions linearly independent in Vif . . . , Vn. On the other hand, the matrix M will contain 272 GROUP CHARACTERISTICS (Ch. XIII just /; linearly independent elements, namely vi, . . . , Vh. Hence by 1, 13G, k^Ji, and h of the multipliers, j8i, . . . , /?* are linearly independent, say /3i, ^2, , /3*. These h expressions can therefore not all vanish unless t>i = zj2 = . . . However, if k>h, the expressions /3i, . . . , /3/, must all vanish if for i;i, . . . , Vn we put, respectively, the conjugate- imaginaries of the characteristics Xi, . . . , x* of the group Gh+i (131, Cor. 1). But these quantities, xT, . . . , Xa, are not all zero (one of them represents the number of vari- ables of Gh+i). We conclude that k>h. Hence, finally, k = h. EXERCISE Prove that if 5i, . . . , Sg are the substitutions of the regular sub- stitution group H, and x<^'), x/^^)^ _ _ _ ^ ^^{k) ^hg characteristics of the transformations corresponding to St in a set of non-equivalent groups into which H breaks up, then these characteristics do not satisfy an equa- tion 0,x/'>+02X*(>+ . . . +CiX|(*> = where the coefficients ai, . . . , at are the same for all the g subscripts /. 138. Theorem 23. No simple group can be of order p'^c^, p and q being different prime numbers * The proof is divided into two parts: 1. li H is the regular substitution group simply isomorphic with a group of order g = p^q^, assumed simple, then one of the non-equivalent transi- tive linear groups G\, . . . , Gt into which H breaks up ( 131) contains (f variables, and one of the conjugate sets of 11 con- tains ^ transformations. 2. Under these conditions an impossible equation is obtained. 1. The relation g = p<'q^ = ni^+n2^+ . . . +nt^ (132, Ex. 3) with the condition that the numbers i, . . . , t are all factors of g ( 135, Cor. 2) and that only one of them is unity ( 132, Ex. 2) implies that at least one of them is greater than unity and prime to />; say nt = q">\. Again, the relation ^ = /'g^ = gi+g2+ . -\-gh with the condition that the numbers g\, - , gn are factors of g and * Bumside, Proceedings of the London Mathematical Society, 1904, p. 388. 138) GROUPS OF ORDER ?>Y 273 that only one of them is unity (as otherwise U would not be simple; cf. 24) implies in the same manner that one of them is a power of />; say gs = P^> 1. 2. We now have a transitive group Gt in ni = (f variables, and a conjugate set oi gs = p^ transformations. Let 5^" denote one of the transformations of this set, x^*^ >ts characteristic, and T the corresponding transformation (substitution) of II. We have ( 135, Cor. 1) where N represents the sum of a finite number of roots of unity. It follows that x^'' =^^', A" being such a sum also ( 116, 7). But, x^*^ is already the sum of ^ roots of unity. Hence, either all these roots are alike, or x^'^ = 0- The first supposition makes S^^ a similarity-transformation, which would be self- conjugate in Gt. This being impossible for a simple group, we infer that x^'^=0; and this not only for the group Gt, but also for every one among the groups Gi, . . . , G* (and their equivalent groups), the number of whose variables, like<, does not contain /> as a factor. Hence, the sum of the characteristics of the transformations corresf>onding to T, from all these groups, is of the form ix">+ . . . +n*x< = l+/>iV''+gx'+ . . . =1+PN", when account is taken of the fact that one of the numbers iti, ...,*, say i, is unity, and that the corresponding character- istic x"^ = 1 However, this sum is equal to the characteristic x(7') in /7, being equal to the sum of the elements of the principal diagonal of the matrix of T. Hence, since x(7 ) =0, But such an equation is impossible by Kronecker's theorem ( 116, 6). EXERCISE Prove that a gioup in which the nunil)er of operators in a conjugate set is the power of a prime niuuber is not simple (Bumside). 274 GROUP CHARACTERISTICS [Oh. XIIJ 139. Theorem 24. A transitive substitution group of degree n and class n 1 contains an invariant subgroup of degree and class n* Proof. 1. Let g be the order of G, and g' =g/n the order of that subgroup G' of G whose substitutions leave fixed a given letter. Again, let H represent the regular substitution group simply isomorphic with G, and W the regular substitu- tion group simply isomorphic with G'. The groups H and H' are of degrees g and g' respectively. Now let H be resolved into its component linear groups, from which we select a set of non-equivalent groups Hi, H2, . . . , Ht, the number of whose variables are respectively Wi = l, W2, . . . , wjfc. Similarly, let H' be resolved into its different linear groups, from which we select a set of non- equivalent groups E'l, H'2, . . , H'l, in respectively n\ = \, n'2, . . . , n'l variables. The latter set and their equivalent groups are all irreducible components of that subgroup of // which corresponds to H'; they are, in fact, contained as irre- ducible components in the subgroups ol Hi, . . . , Ex (and their equivalent groups) which correspond to G' of G. Let us suppose, for any subscript s^k, that the subgroup of H, which corresponds to G' of G breaks up into/,! groups equivalent to H'l, fs2 groups equivalent to H'2, etc. This division may be exhibited clearly by the following equation: 1^. \=UH'i-{-fs2H'2-\- . . . -\-fsiH\, Evidently, | Hi \ =H'i, so that /ii = l, /l2 = 0, . . . ,/l, = 0. Moreover, ( 132, Ex. 3), ni2+22+ . . . +Wfc2=g, n'i2+V+ . . . +?=g'. Again, if ri( = the identity), T2, . . . , Tg> are the trans- formations of \H,\y and Xi( = ), X2*, . . , X/, their char- Frobenius, Sitzungsherichte, etc., 1901, pp. 1223-1225. 139] GROUPS OF DEGREE n AND CLASS n-1 275 acteristics, while 0ie( = '), 02; . . , dg>, are the corre- sponding characteristics of H',, we have {dn = 1) : (3) Xt,=fsi-{-Uet2+ . . . +fada. 2. Assuming for the present that/ii = for a certain sub- script s, the theorem is easily proved. If we denote by Sx the sum of the n characteristics of Us corresponding to the identity and to the substitutions of G of class n,* we have, applying Cor. 2, 128, =i 1-1 ' = y]x-nns, or (4) 2)^ = Ws. Hence, there being w characteristics in Sx, and each being the sum of roots of unity, the equation (4) can be true only if each characteristic x is fis. But then the corresponding trans- formation of Hs must be the identity (1, 1, . . . , 1); and all such transformations correspond to an invariant subgroup of G (cf. 132). This subgroup includes all the substitutions of class n and possibly some more (though not all of G, by virtue of the condition /,i=0). If it includes more, then this new group, of order <g, may be chosen instead of G, and thus we would, by a proof by induction, ultimately show the exist- ence in G of a subgroup of degree and class n, which would obviously be an invariant subgroup of the original group G. There are n 1 substitutions in G which permute all the letters; n(jfl) which permute all but one; and one (identity) which leaves them all fixed. t No two different subgroups of G of class n 1 can have a substitution in common. 276 GROUP CHARACTERISTICS [Ch. XIII 3. Going back to 1, it remains for us to prove that at least one of the numbers /21, /ai, . . , /ti is zero. To accom- plish this we start with the following equations (which we shall establish in 4) : /2i2+/3i2+ . . . +v=i:^' (5) /21/22+/31/32+ . . . +/*i/*2 = ^^'a, ,/ g^ from which we derive Now, since the terms in the left-hand member are positive integers or zero, it follows that one of them is unity and the rest zero; say /22=w'2/21l, /32=W2/31, , fk2=n ifkU Substituting in the second of the equations (5) and sub- tracting n'2 times the first gives us /2i=0, what we set out to prove. 4. To prove (5), consider the sum M = XiXii0u+X,iX2l02+ . . . -^Xv\Xg'\Ba't 4-X,2Xl201t + Xr2X2202l+ +Xp2Xa'20/i + XrtXlk01t + X,iX202<+ . . . -\-Xf>kXg'i,dg't (K^^^O- Let there be h' operators conjugate to T, in G', and there will be h'g/g' conjugates to T in G. Then, adding by columns, we get (135, Ex. 2) Af = /l'(X,iX.l+X,2X,2+ . +Xr*Xrt)^rt=^'^- 1391 GROUPS OF DEGREE n AND CLASS n-1 277 Again, substituting for Xu from (3) and adding by lines we have (131, Cor. 1): t r-1 Hence, equating the two values of M obtained we get (6) {)=^frlfn + e,2^fr2fn+ +e^{-\ + ^fn^) + . . . -\-d^'^frifn=Ai + e,2A2+ . . . +M, say, where the summation extends from r = 1 to r = ^, and 1 <vS.g'. We can now prove the following equation: (7) 0=Ai-q+e,2{A2-qn'2)+ . . . +dMi-qn'i) (l^^g')> where ?=^''''.- When v>\ the equation reduces to the following by'means of (6) : O=-g(l+'20,2+ . . . +n\eu), which is true, since the quantity in the parenthesis is the char- acteristic corresponding to T, in the regular group H' and is therefore zero. If i; = l,the right-hand member of (7) becomes (cf. 1) Ai-q-[-n'2{A2-qn'2)+ . . -\-n'i{Ai-qn'i) =Ai+n'2A2-\- . . . +'^z-5(l+V+ . . . +7) I = 7^ frtifrl +wVr2+ .. ^-n'tfrlj-n't-qg' r-1 (8) =^frtnr-j-,n't. Now, 2^Ii/rr is the number of times the group //'/ (or equivalent groups) enters as a component of the subgroup ] H | of H. In this group, G' will evidently be represented as an 278 GROUP CHARACTERISTICS [Ch. Alll intransitive substitution group made up of g/g' sets of intransi- tivity, of g' letters each. For each such set there are n't groups H't. Accordingly, SI^lJ/rtWr = w'tg/g'. Substituting in (8), the quantity vanishes. Having thus proved (7) for Z' = l, 2, . . . , g', we may apply the proposition stated in the exercise, 137, from which it follows that every coeflScient Apqn'p must vanish. Equating to zero the coefficients for /> = 1 and p = 2, t = l and i = 2, the equations (5) are finally obtained. PART III* APPLICATIONS OF FINITE GROUPS CHAPTER XIV THE GROUP OF AN ALGEBRAIC EQUATION FOR A GIVEN DOMAIN 140. Introduction. The theory of substitutions and groups of substitutions grew out of the investigations by Lagrange, Ruffini and Abel of the question of the solvability by radicals of the general algebraic equation of degree n. We shall answer this question by means of the theory of Galois, which is appli- cable to any algebraic equation, whether its coefficients are con- stants or depend upon one or more variables. In the latter case we must first give a definition of the roots of the equation and the concept equality of two functions of the roots. Con- sequently, we shall begin with the more concrete, and yet typi- cal, case of numerical equations. With a given equation we shall associate a certain group of substitutions on its roots and shall prove that the equation is solvable by radicals if and only if the group is solvable, i.e., if each of the factors of composition of the group is a prime number. If we regard as known not merely the coefficients of the given equation, but also certain constants, such as roots of imity, the solution of our equation may be thereby simpli- fied and the group altered. In fact, most of our concepts, such as the irreducibility of the equation, its group, etc., depend upon the constants regarded as known. In order to specify *This part was written by L. E. Dickson. 279 280 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV these constants briefly and clearly, we shall define and employ the concept " domain." After developing the essential principles of Galois' theory of algebraic equations, we shall apply the theory to various problems in geometry; first to constructions by ruler and com- passes, including the proof of the impossibility of certain con- structions of intrinsic and historic interest; then to the inflexion points on a plane cubic curve, the 27 straight Unes on a cubic surface, and the 28 bitangents to a quartic curve; finally, to a general series of problems on contacts of curves. 141. Number Domains. The set of all rational functions with rational coefficients of the complex (real or imaginary) numbers ^i, k2, . . - , km is called a domain and denoted by R{k\, . . . , kn^- Hence if we perform any one of the four rational operations (addition, subtraction, multiplication, divison by a number not zero) upon any two equal or distinct numbers of the domain, we obtain a number of the domain. We assume that each ki is not zero. The domain contains every rational numbers, since it contains rk\/ki. The domain RiX) is the set of all rational numbers. EXERCISES 1. Every number of R{i), where 1"^= 1, can be given the form a-\-hi, where a and b are rational. Every number of /?(V3) is of the form a -\-bV^. 2. If = -^+V^, then /?() = /?(V^). 3. If =i-|->/2, R{h V2) = /?({). Hint: i-V2=-3A. 4. If k is the real cube root of 2, every number of R{k) can be given the form a-\-bk-\-ck^, where a, b, c are rational. 142. Reducibility and Irreducibility. An integral rational function /(a;) of degree of a variable x whose coefficiehts belong to the domain R is said to be reducible in R if it can be expressed as a product of integral rational functions of x each of degree <n with coefficients in R; irreducible in R if no such factoriza- tion is possible. Example 1. The function *+! is reducible in R{i) since it has the factors xi, but is irreducible in /?(1) and in R{V2). Example 2. x* + l is reducible in any domain which contains either y/i, or V^, or i=Vl, or t=il-{-i)/'s/2, but is irreducible in all other 143] REDUCIBLE AND IRREDUCIBLE FUNCTIONS 281 domains. Indeed, its linear factors arc r, .r~'; while every quad- ratic factor is of one of the forms x'i, x^+axl (*= 2). Example 3. x' 2 is irreducible in ^(1). For, if it were reducible, it would have a linear factor xa/b, where a and b are relatively prime integers, of which b may be taken to be positive. Then a' 26' = 0. If b has a prime factor 0{ii> 1), then /3 divides o* and hence divides a, whereas a and b have no common factor /3. Thus b=l, o*=2. Hence the positive integer a divides 2, so that a=\ or 2, and a'= 1 or 8, whereas a= 2. Hence x 2 is irreducible in R(l). li f{x) is reducible in R,f{x)=0 is called a reducible equa- tion in R; in the contrary case, an irreducible equation in R. 143. Irreducible Binomials. // p is a prime number and if A is a number of a domain R, but A is not the pth power of a number of R, then x^A is irreducible in R. The roots oi x'' = A may be denoted by r, or, coV, . . . , co^'^r, where is an imaginary pth. root of unity. Let there be a factor with coefl5cients in R oi x^A. It has a constant term of the form zLosY, where 0<t<p. By the theory of numbers, there exist integers b, c such that btcp = \. Hence R contains (wV)* = w*V^+^ = oi'^rA'^r'A', where r' is one of the above roots. Thus r' is in R, so that A is the pth power of a number r' in i?, contrary to hypothesis. 144. Theorem. Let the coefficients of the polynomials f(x) and g{x) be numbers of a domain R and let f(x) be irreducible in R. If one root a of f(x)=0 satisfies g{x)=0, every root of f(x) =0 satisfies g{x)=0 and f{x) is a divisor of g{x) in R. The greatest common divisor h(x) of f{x) and g{x) is not a constant, since it has the factor xa. The usual process for finding h(x) involves only rational operations; hence its coef- ficients are numbers of the domain R. Since f{x) is irreducible in Rf its divisor h{x), with coefhcients in R, is of the same degree as f{x), and hence equals cf(x), where c is a number in R. But h{x) divides g(x). Hence /(x) divides g(x) in R. 282 GROUP OF AN ALGEBUAIC EQUATION (Cu. XIV EXERCISES 1. If x'+cx^+dx+c=0, where r, d and e are integers, has a rational root, that root is an integer. Hint: Let x=a/b, where a and b are relatively prime integers, and multiply the equation obtained from the cubic equation hyb*. 2. An integral root of the equation in Ex. 1 is a divisor of e. Hint: It divides x^, cx^ and dx. 3. x' 3x4-1 is irreducible in R{i). 4. X* 7x+7 is irreducible in R(l). 5. State and prove for equations of degree n the theorems correspond- ing to those of Exs. 1, 2 for = 3. 6. x<+x'+x2+x+l = is irreducible in R(\). Hints: It has no rational root (Ex. 5). If it has the factors x-'\-ax+r,^x^+bx+r~ ^, where o, b, r are rational, then 0+6=1, o6+r+r-=l, ar-^+br=l. Either a = ^{lVE), 6 = K1tV5), r=l; r , I or o= , b= , r*+r^+r^+r-\-l = 0. r+1 r + l 145. Functions with n\ values. Let R he a. given domain which contains all of the coefficients of a given numerical equation (1) /(a;)^x-Cix"-^+C2:c-*- . . . +(-l)% = 0, which, without * real loss of generality, will be assumed to have the distinct roots xi, . . . , ocn- There exist integers wi, . . . , Wn such that Vi = miXi-\-m2X2+ . . . +mrtXn gives rise to nl numerically distinct functions Vs when the ! substitutions s on xi, . . . , Xn are applied to it. For, if 5 and s' are different substitutions, F, and Vs- are not equal /\ , identically as to mi, ... , nin. We can, however, choose integers Wi, . . . , Wn which satisfy no one of the !(! 1)/2 equations of the form Vs = Vs'. In fact, mi=tn2 is the only one of these equations involving only mi and m2. Give to nil any integral value (say 0) and to W2 any integral value * For, if it has a multiple root, fix) and its derivative fix) have a greatest common divisor gix) with coefficients in R. Then f{x)/gix) has its coefficients in /?, has no multiple root, and vanishes for each root of /(x)=0. 1461 GALOISIAN RESOLVENTS 283 9^mi (say 1). Consider the equations involving ma, but not nu (J>3); they determine certain values of wa; give to mz any integral value distinct from the latter. Next we give to Mi an integral value distinct from the values of W4 deter- mined by the equations involving m^, but not mi(i> 4) ; etc. For mi=0, mj= 1, the values of mt to be avoided are * 1 1 x-1 X 0, 1, X, 1-X, -, , ^ , ^_^, where Xi Xj Xl X3 Thus Xi+mxi has six distinct values under the six substitutions 1, a=(xiXi), b=(xiX3), c={x2X3), d={,XiX2X3), e={xiXiXj), if m is an integer distinct from the above eight numbers. We shall often employ as an example the equation (2) x+x^+x+l = 0, with the roots x,= 1, X2=i=V-l, X3= i. Here X= i, so that xj+wx, is six-valued if iht^O, 1, j, \.i, \{\zht), and hence if m=l. The six distinct values of x^Xi are Fi=X2 xi=l+i, Vi,=XiX3=2i, Vc = XiXi=\i, Va=-V Vi=-V, Ve=-Vc. 146. Galoisian Resolvents. The substitutions on a;i, . . . , Xn will be denoted by Si, . . . , 5!, where ^i is the identity. If SjSt = Si, and we apply Sj to Vi and then St to the resulting function F, , we get Vs^. When k is fixed, buty takes the values 1, 2, . . . , !, then / takes the same values in some new order. Hence St merely permutes V,, F,,, . . . , V. amongst themselves. Thus the elementary symmetric func- tions of these F's are symmetric functions oi xi, . . . , Xn, If one of the cross-ratios of four points is X, the others are 1 X, etc. The six transformations X'=X, X'=l-X, .... X'=X/(X-1) form a group. Cf. Ex. 3, 6 and Ex. 7, 87. 284 GROUP OF AN AI.(JKRHAFC EQUATION [Ch. XIV and hence * arc integral rational functions of wi, . . . , m, Ci . . . , Cn with integral coeflicicnts, and therefore are num- bers of the domain R. Thus the coefiicients of the polynomial in V, given by the expansion of (3) F(V)^iV-VS{V-Vs) . . . (V-VsJ, are numbers of the domain R. If F{V) is reducible in R, let G(V) be that irreducible factor in R for which G(Fi)=0. If F{V) is irreducible in R, take G{V) to be F{V) itself. In either case, G(F)=0 is an irreducible equation in R, having the root Vi; it is called a Galoisian resolvent of equation (l) for the domain R. The corresponding resolvent for equation (2) in R{\) is G{V)^{V-V,){V-Vc) = V^-2V-\-2 = Q. For the domain R{i), the resolvent is F Fi = 0. 147. Theorem. Let <t>{xi, . . . , Xn) be any rational integral function, with coefficients in a domain R, of the roots of an equa- tion with coefficients in R. Let s he any substitution on the roots and let it replace <i> by 4>s, and an nl-valued linear function Fi, with coefficients in R, by F,. Then (4) 0,= F'iVsY where X is a polynomial with coefficients in R, while F' is the derivative of the polynomial (3) with coefficients in R, so that F'(Vs) 7^0. Thus </)s is the same rational function p(Va) of V, that <t> = <t>i is of V\: If S}St = si, then Sk replaces <^s^ by </),,j. Thus St permutes <^i, . . . , <^,^, in the same manner that it permutes Fi, . . . , F,^, ( 146). Hence the terms of (r,\ xm-^ ^^^^^ +^ _W). , . . F{V) (.3) Mn=<^i^73y^+<^.',prrir+ -^K.v-v.^, * Several detailed proofs of this fundamental theorem on symmetric functions, which is frequently applied below, are given in Dickson's Eicmcntary Theory of Equations. 147] GALOISIAN RESOLVENTS 286 are merely permuted amongst themselves by any substitu- tion on a:i, . . . , Xn. Thus the coefficients of the polynomial X(F) are rational integral symmetric functions oi xi, . . . , Xn with coefficients in the domain R, and hence equal numbers of R. Taking V=V we obtain X(F,) = 0,F'(F,). Since F'(F,)?^0, weget (4). Example. Recurring to the special equation (2), we shall obtain the explicit expressions (4) for the case = X2, Fi=X2 jci. Then (3') F(F) = F+4F<+4P^2 + 16, (50 x(F)=F(F)|-^^+^^+-^^+-^^+^^+-^^ ^ '\V-V, F + F. V-Vb V-Vc V+Vi, V+Vc = -2F5-4F*-12F3-8F2-16F-48, as shown by inserting the values of ^i, . . . , Vc given at the end of 145. Hence /T. N -2FiS-4Fi*-12Fi3-8Fi2-16Fi-48 Xt=p{Vi) = . 6Fi*+16Fi3+8Fi In view of the theorem, we have Xi=p{Va), X2=p{Vb), X3=p{Vc), X3=p{Va), Xi=p(Ve). These results may be verified by evaluating the expressions. . The numerator and denominator of the above fraction for X2 may be expressed as linear functions of Vi by means of the relation Fi* 2Fi4-2=0 of 146. We get -48Fi+32 _ (-3Fi+2)(Fi+2) _ -.3Fi^-4Fi+4 _ -10Fi + 10 , 16Fi-64 ~ (Fi-4)(Fi+2) ~ F,2-2Fi-8 ~ -10 ~ ' While therefore X2 is numerically equal to Fi 1 (each being t), it is not admissible to take this reduced function r(Fi) = Fi 1 as the function p(Fi) of the theorem, since it would no longer be true that, by applying the substitution a'={xiX^, we would have xi = r{Va)- Indeed, if we apply a to Vi \=X2Xi\, we obtain Xi.X2\9^Xi. The explanation is that we should reduce the second member of the true relation Xi = p(Fa) by means of F*o+2Fa+2=0; we thus obtain 16Fa+64 which is the correct value of X\. Since Vc satisfies the same quadratic equation as Vi, our first reduction yields also the true relation XtVcl. 286 GROUP OF AN ALGEBRAIC EQUATION [Cu. XIV We therefore see why we obtain a true relation if we apply the substitu- tion c to the members of Lhe reduced relation X2= Ki 1, and why it would be accidental if we obtained a true relation when we apply to X2=Ki 1 any substitution other than c and the identity substitution. This example brings us to the core of our subject and indicates the care which must be taken in its development. 148. The Group of an Equation for a Domain R. Let the roots of a Galoisian resolvent G{V)=0 of the given equation be (6) Fi, Va, V,, . . . , F in which the subscripts denote the substitutions on :ri, . . . , Xn by which these F's are derived from Vi. We shall prove the Theorem. The g substitutions (7) 1, a,b, . . . ,1 form a group G, called the group of the given equation (1) for the domain R. We are to prove that the product rs of any two of the sub- stitutions (7) is one of those substitutions. Take F, as the function in 147. Then where X is a polynomial with coefficients in R. Since F, is a rootof G(F)=0, X(F) = '(F); is satisfied when F=Fi. Multiplying the left member by the gth power of F'{V), we obtain an integral function /(F) of V which vanishes for F = Fi and has numbers of R as coefficients. Hence the root Fj of the irreducible equation G(F)=0 in R is a root of /(F) =0 ( 144). Since F'{Vs)7^0f we may divide /(F,) by the gth. power of F'{Vs) and get '=K^)=^(^"^- Hence Vn is one of the functions (6) and rs one of the substi- tutions (7). 149) PROPERTIES A, B OF THE GROUP 287 Example. For the domain of rational numbers, a Galoisian resolvent of the cubic equation (2) was seen in 140 to have the roots Vi and Ve. Hence the group of (2) for R{1) is {l, (a:2a;3)|. But for the domain R(i), a resolvent is F Fi = and the group is the identity. 149. Characteristic Properties of the Group G of a Given Equation for a Domain R. Let ^/^ be the quotient of two rational integral functions of the roots with coeflScients in R, such that ^?^0. We have (4) and ^' F'iVsY where m is a polynomial with coefficients in R, and fi(Vi)9^0. If 5 is a substitution of G, then xJ/st^O. For, if )u(F) =0 has the root Vs, it has also the root Vi in common with the irreducible Galoisian resolvent G{V)=0 ( 144). Hence the functions <f>, \{Vs) ^s niVs) is = l,a, ...,/) are defined for each substitution s of the group G. Suppose that these g functions are equal numerically, in other words, that 0/^ is unaltered in value by all of the substitutions of G. Then </>_l[X(Fi) \{Va) ,X(F01 The second member is a rational symmetric function, with coefficients in R, of the roots (6) of G(F) =0 and hence equals a rational function of its coefficients, which belong to R. H[ence 0/^ equals a number in R. A. If a rational function with coefficients in R of the roots of an equation with coefficients in R remains unaltered in value by all of the substitutions of the group G of the equation for R, it equals a number in R. B. Conversely, if a rational function of the roots with coef- ficients in R equals a number in R, it remains unaltered in value by all of the substitutions of G. It remains to prove B. Let <t>/4/ equal the number r in R. Then \iV)/fi{V)-r vanishes for V=Vi. Hence the equation 288 GROUP OP AN ALGEBRAIC EQUATION [Ch. XIV X(V') f/u(V)=0 with coefficients in R is satisfied by every root V, of the irreducible equation G{V) =0 ( 144). Hence so that </>/^ is unaltered in value by the substitutions of G. Example. Consider a cubic equation, like (2), with a rational root Xi and no multiple root. By property B with Xi as the rational function, its group for any domain containing the coefficients has no substitution other than 1 and (xiXs). If the domain contains Xn and hence also Xj, the group is the identity; this is the case with equation (2) for R{i). In the contrary case, there must, by property A, be a substitution altering Xj, so that the group is |l, (x2X3){. Since an !-valued function Vi with coefficients in a given domain R can be chosen in an infinitude of ways, there are infinitely many Galois resolvents G{V)=0. Our definition of the group G of the given equation for the domain was based upon a single such resolvent, i.e., upon a particular Fi. It is a fundamental proposition that different functions Vi always lead to the same group G. This follows from the Theorem. The group of a given equation for a given domain R is uniquely defined by properties A and B. First, suppose that G' = \l, a', b', . . . ^ m'\ is a group for which property A holds. Then the coefficients of <1>{V)^{V-V{){V-V^){V-V,) . . . (F-F,), beiilg symmetric functions of Fi, . . . , V,^, are unaltered numerically by the substitutions of G' and hence equal num- bers in R. Since the equation 0(F)=O, with coefficients in R, admits one root Vi of the irreducible Galoisian resolvent G(F)=0, it admits all of the roots (6) of the latter ( 144). Hence 1, a, . , . , / occur among the substitutions of G\ so that G is a subgroup * of G' . Second, suppose that r = {l, a, /3, . . . , x} is a group for which property B holds. Then the Galoisian function * In Part III, a group is included among its subgroups. 150) TRANSITIVE GROUP . 289 G{Vi), being equal to the number zero in R, remains unaltered in value by the substitutions a, 0, . . . , x, so that = G(Fi)=G(O= . . . =G(F,). Hence Va, . . . , V^ occur among the roots (6) of G{V)=0. Thus r is a subgroup of G. liG' = T, then G' = G. In view of its repeated application below, we state our second result as the Corollary. // every rational function of the roots with coefficients in R which equals a quantity in R is unaltered in value by every substitution of a group T, then T is a subgroup of the group G for R of the equation. 160. Transitive Group. We shall make much use of the Theorem. // an equation is irreducible in a domain R, its group for R is transitive, and conversely. Consider an equation f{x)=0 irreducible in R. Contrary to the theorem, suppose that its group G for R is intransitive and contains substitutions replacing Xi by iCi, 0:2, . , Xm, but none replacing xi by one of Xm+i, . . . , Xn. Hence every substitution of G permutes xi, . . . , Xm amongst them- selves and thus leaves unaltered any symmetric function of them. Hence g{x) = (x-Xi)(x-X2) . . . (X-X,n) has its coefficients in R, in view of property A. Thus f(x) has the factor g(x) in R, contrary to its irreducibility in R. To prove the converse, let G be transitive and the equation f{x)=0 be reducible in R. Let the preceding function g(x) be a factor of f{x), the Coefficients of g{x) being in R and its degree m being less than n. Since g{xi) equals the number zero of R, it is unaltered by every substitution of G (property B). Since G is transitive, g{xi)=0 for t = l, . . . , . This tontradicts m<n. Example 1, Find the group G of x 7x-|-7 = for /?(1). The equation is irreducible (Ex. 4, 144), so that G is transitive. It 290 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV will therefore be the alternating group G={l, (xix^xt), (xix^i)] if shown not to be the symmetric group. The square of the function (8) , ^ = {xi -Xi)ixt Xt) (xj - xi) is the discriminant * 49 of our cubic, so that \f/ equals a number 7 of the domain R{1). A transposition changes ^ to ^^ and hence is not in G (property B). Hence G^Gz- ExABiPLE 2. Find the group G oi x*+l = iov R{1). II Xi, . . . , Xi are the roots of x* +ax^ +bx^ +cx+d=0, then yi=xiXo-\-X3Xi, y2=xiX3+xtXi, yi=XiXi+XiXt are the roots of the resolvent cubic equation (9) y^-by^+{ac-id)y-a^d+4bd-ci=0, as shown f by finding yi+yi+yz, - , yiy2yi, or by Ferrari's method of solving the quartic equation. For x< + l = 0, (9) is y 4y=0 and has three distinct rational roots. Hence (by B), each substitution of G leaves yi, yt and yz formally unaltered. Now yi is unaltered only by the substitutions of the group (10) G8=|l, (12), (34), (12)(34), (13)(24), (14)(23), (1324), (1423)}, while yz is unaltered only by the substitutions of (11) C'8=}1, (13), (24), (13)(24), (12)(34), (14)(23), (1234), (1432)}. The substitutions common to these groups form the group (12) G,= {1, (12)(34), (13)(24), (14)(23)}. By Ex. 2 of 1 142, ac* + l is irreducible in i2(l). Hence G=Gi. Example 3. Find the group G of x<+x'+a;2 4-x+l = for /?(1). The roots may be denoted by Xi = t, X2=e^, Xi = t*, Xi=e*, where c is an imaginary fifth root of unity. Then ^2 = 2, while yi = '+ and y=t*+( are the roots i( liVs) of y^+y 1 = 0, as may also be shown by use of (9). Thus G is a subgroup of G's, which leaves ya formally unal- tered. But the substitutions (13), (24), (12)(34) and (14)(23) replace Xj Xi*, whose value is zero, by functions which are not zero. Hence G is a subgroup of the cyclic group generated by (1234). Since the equa- tion is irreducible in R(l), by Ex. 6, 144, G is this cyclic group. *For X* + px-\-q= 0, if'^=-4p*- 27 qK t Cf . Dickson's Elementary Theory of Equations, p. 39, 3. 1501 TRANSITIVE GROUP 291 EXERCISES For the domain of rational numbers, find the group of 1. x-l = 0. 2. (x-I)(x + l)(:c-2) = 0. 3. x-9x+9 = [compute (8)]. 4. x-2 = 0. 5. For the domain /?(&;), where w is an imaginary cube root of tinity, the group of x' 2 = is of order 3. [Compute (8)]. 6. For the domain RU), the group of x< + l = is of order 2. 7. Find the group G of a reciprocal quartic equation X* +ax^ +bx^ +ax +1 = for the domain R=R{a, b), when it is irreducible in R. Hints: Choose the notation for the roots so that Xiaf2=a;3a:4=l. Then one root of the cubic (9) is ^1 = 2; thus y^ and yz are the roots ^blVA of y*4-(2-6)3'4-a*-26=0, where A = (^b+iy-a^-. The three y's are distinct; for example, yiyi=(xiX4)ix3X2). Hence G is a subgroup of Gg, given by (10). Further, G is Gt, given by (12), if and only if Va isia R. By the usual substitution v=x+l/x, our quartic becomes v*+av+b2 =0, whose roots are therefore iCi+X2 and X3+x^. Its discriminant B=a^4{b2) is thus the square of t=Xi+XiX3Xi. Why is MO? Now (1324), (14) (23) and (13) (24) replace /by -t, while the first four substitutions in (10) leave / unaltered. Again, ytyz is unaltered only by the subgroup G* of Gs, being changed in sign by the remaining four substitutions of d. Hence {(yiys) is unaltered only by the subgroup Ci generated by (1324). Hence G=C^ if and only if VaB is in R. If a transitive subgroup of Gs does not contain (1324) or its inverse (1423), it contains (13)(24) and (14)(23), the only remaining substitutions replacing Xi by Xj and Xt, respectively, and hence is G*. Thus if G is not Ci or G4, it is Gs. It follows by formal logic that G=G8 if and only if neither y/A nor VaB is in R. 8. If the quartic in Ex. 7 is reducible in R, it is the product of two fac- tors x*-\-px+r and x^+qx+1/r, where p, q, r are in R, and P 1 p+q=a, -+rq=a, r-^ \-pq=o. r r If r=l, p and q are in R only when Vb is in R. If r= 1, then a = and y/b2 must be in R. If rVl, we may eliminate p and q and obtain for y=r+l/r the quadratic * in Ex. 7 with the roots yt, y,; thus R must contain V]4 and the square roots of (Jft 1V^)* 4, the latter being the values of (r l/r)*=y2 4 for y=yj, yj. By Ex. 7, ABy^O if the four roots are distinct. 9. If a=b=0, then A = l, 5 = 8, and x* + l is irreducible in RiD, and * Except for a=0; then P=q0, y=b. 292 GROUP OF AN ALGEBRAIC EQUATION ICn. XIV has the group G4. If u = 6= 1, then .4=5/4. B=5, anda;+x+a;*+x + l = is irreducible in /?(1) and has the group d. Let /(ac) = aox"+flix"~ '+. . . =0 be an equation with rational coef- ficients, irreducible in the domain of rational numbers. Prove * Exs. 10-14: 10. If there is a complex root of absolute value unity, the equation is a reciprocal equation of even degree. 11. If there is a root r+si, where r is rational, then n is even and the n roots may be paired so that the sum of the two of any pair is 2r, whence r= fli/(nflo). In particular, if r=0, the equation involves only even powers of x. 12. If there is an imaginary root a+bi whose norm a^-\-b^ is rational, then n is even and the n roots can be paired so that the product of any two of a pair is o*+6^ 13. If there is a root whose absolute value p is rational, p can be expressed in terms of the coefficients. (p**=a/oo.) 14. If we set x=py in the equation in Ex. 13, we obtain a reciprocal equation in y. Equations whose Coefficients Involve Variables, 151-6 151. Definition of the Roots. We begin with the so-called general equation (1) whose coefficients ci, . . . , Cn are independent complex vari- ables. Let a(ci, . . . , Cn) be its discriminant. Let Oi, . . . , an and Ai, . . . , ^n be any two sets of constant values of ci, . . . , c for which A?^0. We shall prove that the A's can be derived from the o's by con- tinuous variation such that, for each intermediate set of values, Ay^O; expressed in geometrical language, there is a continuous path from the point (a) to the point (A) not passing through a point of the locus A = 0. We shall prove this by induction from nl to n, assuming that, if P(ci, . . . , Cn-i) is any pKjlynomial in Ci, . . . , Cn-i, zero neither at (/i, . . . , In-i) nor at (1, . . . , Ln-i), there is a continuous path from (/) to (L) not passing through a point of the locus P=0. Neither of the polynomials A(a, . . . , C-i, On) A(Ci, . . . ,Cn-u An) is identically zero, since the first is not zero when each ' = a<, and the second is not zero when each Ci = Ai. Hence there are constants o< for which A(ai, . . . , an-i, Un)5^0, A(ai, . . . , a-i, An)7^0. Thus there is, by hypothesis, a continuous path from (ci, . . . , On-i, a) to (ai, . . . , on-i. On), not passing through a point of P=A(ci, . . . , Cm-\, c) = and composed only of points with c = a, and hence not passing Exs. 10, 11, 13 are due to Dr. A. J. Kempner. 1521 FUNCTION DOMAINS, EQUALITY 23 through a point of A(ci, . . . , Cn) = 0. The equation A(ai om-uv) = in r has only a finite number of roots. Hence v can be varied contin- uously from On to An SO that A(ai, . . . , an - 1, v) 9^0 at each intermediate point. The combination of our two paths gives a continuous path from (o) to (ai, . . . ,an-i, An) not passing through a point of A = 0. Similarly, there exist constants /3j for which A03i, . . . ,j3-:, an-i, An)7^0, A(fii, . . . , /3-,, An-i, ^)70. By hypothesis, there is a continuous path frcm (ai, . . . ,_!, An) to (/3i, . . . , A,-s, a-i, An), not passing through a point of a(ci, . . . , Cn-i, an-i, An) = and composed of points with Cn-i=a_i, Cn=An, and hence not passing through a point of A(ci, . . . , <^) = 0. Evidently there is a continuous path from our final point to E={0i, . . . , /3_i, An -I, An) not passing through a point of A = 0. We now have a continu- ous path from (a) to E not passing through a point of A=0. Proceeding in this manner, we finally get such a path from (a) to (^4). Let Xi", . . . , ocn" he the n distinct roots of the equation with the coeflBcients ai, . . . , On- These roots receive increments as small in absolute value as we please when Oi, . . . , On are given increments suf- ficiently small in absolute value.* Hence if we proceed along our path from (a) to (^), we obtain a definite coordination of the roots x'l, . . . , x'n of the equation having the coefficients Ai, . . . , An with the initial values Xi", . . . , Xb". Thus the latter and definite paths radiating from (a) lead to n functions Xi, . . . , x of Ci, . . . , Cn uniquely defined for every set of c's for which A^^O, and called the roots of the general equation (1). In fact, for a particular set of c's, the roots of the equation are the values of the functions Xi, . . . , Xn for those c's. Our main investigation is the comparison of a rational function of the roots with that derived by a substitution on the roots; hence we shall not be interested in values of the c's for which A=0, i.e., for which two or more roots become equal. The same scheme defines a fortiori the roots of any equation whose coeflScients are functions of one or more variables. We retain only those sets (A) which are sets of values of our present coefficients. The fact that certain of the sets intermediate to (a) and (^4) are not now values of our coeflBcients does not disturb the coordination of the roots at (A) with those at (a). The scheme therefore assembles the root values into root functions. 152. Function Domains, Equality, Group of an Equation. Instead of a domain composed of constants, we now employ a domain R{ki, . . . , i^) composed of all rational functions *The roots are continuous functions of the coefficients. For a proof, see Weber's Algebra, vol. 1, 1895, p. 132. 294 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV with rational coefficients of ^i, . . . , ^, either all of which are given functions, or certain of which are functions and the others are constants. For example, -R(V3, k) is composed of all rational functions of the variable k with coefficients of the form a+ftVs, where a and h are rational numbers. Two polynomials and ^ in the variables V\, . . . , Vi with coefficients in R are called equal if they have the same numerical value for every set of numerical values which v\, . . . , vi, ki, . . . , km can assume. For example, if vi and V2 are the roots of x'^-\-kx-\-l=0, then t'i+z'2= ^z'i2^2- The equality of two rational functions of z^i, . . . , z^i is defined similarly, but with restriction to those sets of values of the v^s and ^'s for which each denominator is not zero. If \{/ is derived from (/> by a substitution s on Vi, . . . , Vi and if ^ = <^ in the present sense of equality, we shall say that <j> is unaltered by s. The definitions and theorems in 142-4 concerning irre- ducibility evidently hold for the present generalized domain R. Each element (function) in R is conveniently called a quan- tity in R. Proceeding as in 145-9, we see that an equation whose coefficients ci, . . . , c, are any given functions or constants has a definite group G for any given domain containing ci, . . . , Co- 153. Group of the General Equation. Let the coefficients Ci, . . . , Co be independent complex variables. Let Xi, . . . , Xn be the roots in the sense of 151. Lemma. // a rational integral function ^{xi, . . . , Xn) of the roots with constant coefficients is zero for every set of values of c\, . . . , Cn, each coefficient of \[/ is zero. We consider those sets of values of ci, C2, . . . , c which are the values of the elementary symmetric functions Sz;i, 2z'i2;2, . . . , Vi . . . Vn of the independent variables vi, . . . , Vn- For each set of values of the v's we therefore obtain a set of values oi xi, . . . , Xn forming a permutation of the v's. Consider the product P of \l/{vi, . . . , ^) by the func- tions obtained from it by applying the various substitutions, 153] GROUP OF THE GENERAL EQUATION 295 other than identity, on i^i, . . . , ti.. For every set of t>*s, one factor of P is a value of \l/(xi, . . . , Xn) and hence is zero. Thus P is zero identically in the v's. Hence some factor ^ of it is zero identically. Theorem. The group of the general equation for the domain R defined by its coefficients and any chosen constants is the sym- metric group. The coefl&cients of its Galoisian resolvent G{V)=0 are rational integral functions of ci, . . . , c with constant coef- ficients. Replace Ci, . . . , c by the elementary symmetric functions of xi, . \ . , Xn. Then G{V) becomes a poly- nomial P{V) whose coefficients are rational integral functions of the x's with constant coefficients. Let Vi = I,miXt, where mi, . . . , mn are distinct integers, be the function used in constructing G{V). Then PiVi) is a rational integral func- tion of the x^s with constant coefficients which is zero for every set of values of the c's. By the Lemma, P{Vi) is zero identi- cally in the x's. The function derived from it by applying any substitution 5 on the x's is therefore zero identically in the x's. Since the coefficients of G{V) are unaltered by this substitution, we get G(Vs)=0. Hence every substitution 5 occurs in the group of the equation. EXERCISES 1. Prove the last theorem by showing that properties A and B in 149 hold when G is the symmetric group. Note that il 4t=(t>s for all values of the c's, then, by the Lemma, (f>=<i>s identically in the x's; if this is true for every substitution s, <t> is symmetric and hence is a rational function with rational coefficients of Ci, . . . , Cn and the coefficients of <f>. Next, if <t>ixi, . . . , Xn) equals a rational function of the c's and hence a rational symmetric function ^ of the x's, for every set of c's, then ^s ^ identically in the x's (Lemma), so that <A is symmetric, and property B holds. 2. If Gis the group of /(x, c)=x" cix~^-f- . . . Cn = Ofor the domain R=R(ci, . . . , Cn, ki, . . . , ki), where ki, . . . , ki are constants, and if c'l, . . . , c'n are values which ci, . . . , c^ can take, then the group G' ol fix, c') = for R'=R(c'i, . . . , c'n, ki, . . . , h) is G or a subgroup of C. Hint: If G(V,,c) = is the Galoisian resolvent of /(x, c) = for R, the group of /(x, c') = for R' is G or a subgroup according as C(K, c') is irre- ducible or reducible in R'. 296 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV 3. Hence the group of the general cubic x*cix*+cixci=0 for Rict, Ci, Ci) is Gt, since that of x' 2 = for R{\) is G. 4. For a and b independent variables, the group of x* +fl* + JaC +ox +1=0 is G. 5. The group of an irreducible quartic equation for a domain/? is the symmetric group if no one of the roots yi, y2, yz of the resolvent cubic equation (9) is in R, and if the product P of the differences of the y's is not in R. Hints: Its group G is not (10) or (11), and not the group G"g leaving yj unaltered, nor their common subgroup (12), nor ?. cyclic group of order four, necessarily contained in one of these groups of order eight. The only remaining transitive groups are the alternating and symmetric groups, 16. But G is not the former in view of P. 6. The group of an irreducible quartic for R is Gn if that of the resol- vent cubic (9) is Gt. 164. Rational Functions belonging to a Group. Let 1, a,b, . . . , ^ be all of the substitutions of the group (j of a given equation for a given domain R which leave unaltered (in the sense of 152) a given rational function ^{xi, . . . , Xn) oi the roots with coefficients in R. Since \p \f/a has the value zero, we have xh> ^<a = by property B. Hence \t/ab = ^ and ab is in the set 1, a, . . . , ^, which therefore forms a group H. We shall say that ^ belongs to the subgroup H of G. Example. Let Xi and 'a;2= a;i be two roots of r;<+l = 0. The only substitutions of its group (12) for R{1) which leave xr unaltered are 1, (12) (34). Hence they form the subgroup to which Xi^ belongs. Conversely, let H be any given subgroup of G. Let Vi be any !-valued function of the roots with coefficients in R, and let Vi, Va, . . . , Vt be the functions obtained from Vi by applying the substitutions of H. If p is a suitably chosen number in i?, ^=(p-Fi)(p-F) . . . (p-n) is a rational function of the roots Xi with coefficients in R which belongs to H. For, if 5 be any substitution of H, then s, as, . . . , ks are distinct and are in H, and hence form a permuta- tion of 1, a, . . . , ^. Thus ^ equals ^.=(p-F,)(p-F) . . . (p-FJ. 154] RATIONAL FUNCTIONS BELONGING TO A GROUP 297 But, if 5 is a substitution of G not in H, then ^, is not identical with ^ as to the variable p, since V, is different from Vi, Va, . . . , Vt. We may therefore choose an integer p such that 4^ belongs to H. This proves Theorem 1. Every rational function ^ with coefficients in R of the roots of an equation with the group G for the domain R belongs to a definite subgroup of G. There exist such functions ^ belonging to any assigned subgroup of G. We next prove the important supplementary Theorem 2. // a rational function \f^, with coefficients in a domain R, of the roots of an equation with the group G for R, be- longs to a subgroup H of index v under G, then the substitutions of G replace ^ by exactly v distinct functions; they are the roots of an equation (13) g{y)^iy-^i){y->p2) . . . (3'-^.) with coefficients in R and irreducible in R. As in 10, let (14) G = H+Hg2+Hg^+ . . . +Hg,. Let h be any substitution of H. Then ^A(r, = ( ^^^)g^ = (rA)*/, = 4^9^- Thus \p takes at most v values under G. But, if ht = hj U<i), then 4^g^g-i = y^^ so that gtgj'^ is a substitution h oi G leaving ^ unaltered and hence is in H. Then gi = hgj, contrary to (14). Thus 4^1, ^2, . ' . , ^y are distinct, where ^ has been written lor ^fff. They are called the conjugates to ^ = ^i under G. Any substitution s ol G _ merely permutes ^i, . . . , ^^, amongst themselves. For, any product gtS may be written in the form hgj where A is in ^; then Ms = ^g^ = fhtj = hj = h- Hence the coefficients of (13) are unaltered by every sub- stitution of G and therefore equal quantities in R. If giy] has a factor with coefficients in R, it has a factor 298 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV y{y) which Is zero for y = \J/i and hence (by B of 149) for y = ^2, . . . I y = ^,- Thus y=g, so that g{y) is irreducible inR. Example 1. The group G of x'+x^-\-x+l = for R{1) is {l, (ocxj)}, if xi= 1 denotes the real root. The conjugates to \//i = X2xi under G are lAi and i2=XiXi; they are the roots of y'* 2y+2 = 0. Example 2. The group G of x*+l = for R{i) is {l, (xiXj)(x2iC4)} if a:i = , ici=i, xs= , a;4= le, where =(l+i)/'^, so that t''=i. The conjugates Xi and xz to xi under G are the roots of y^ i=0, which is irre- ducible in R{i), to which e does not belong. 155. Galois' Generalization of Lagrange's Theorem. // a rational function <j), with coefficients in a domain R, of the roots of an equation f{x) = 0, with the group G for R, remains unaltered by all those substitutions of G which leave unaltered another rational function \f/ of the roots with coefficients in R, then (f> equals a rational function of rj/ with coefficients in R. In case ^ is an !-valued function Vi, the only substitution leaving ^ unaltered is the identity, and this leaves any <j) unaltered. For this case, the theorem states that any rational function <i> with coefficients in R equals a rational function of Vi with coefficients in R. This follows from the like result in 147 for the rational integral numerator and denominator of <t>. Let H be the subgroup of G of index p to which ^ belongs. By means of (14), we obtain the v distinct conjugates ^i, . . . , ^, to ^=^1 under G. Since every substitution h oi H leaves unaltered, each product hgt replaces <j) by (}H = (t>g^. Any sub- stitution 5 of G replaces ^< by a certain rj/j (end of 154) and hkewise <fH by (f>j. Thus, for ^(3;) defined by (13), xW-g(y)(-^+-^+ . . . +-i~r) \y-tj/i y-}p2 y WvJ is an integral function of y each of whose coefficients is unaltered by every substitution of G and hence is in R. Taking ^1 = ^ as y, we get <^ = X(^)H-g'(^). The theorem will be shown to be a generalization of Lagrange's Theorem. If a rational function <f) of the independent variables x\, . . . , Xn remains unaltered by all those substitutions on Xi, . . . , Xn which leave unaltered another 156] ADJUNCTION TO THE DOMAIN 299 rational function \f/ of xi, . . . , Xh, then <t> equals a rational function of \l/ and the elementary symmetric functions Cl=TtXi, . . . , Cn=XiX2 . . . Xn. The group of the equation with the coefficients ci, . . . , Cn for the domain R, defined by ci, . . . , Cn and given constants ki, is the symmetric group. For, property A then states that any symmetric function of Xi, . . . , Xn with coefficients rational in the ^'s is in R, and is the well-known theorem on symmetric functions. Conversely, a function equal to a quan- tity in R is symmetric. Example. y2=xiX3+a:2:j;4 is unaltered by all of the substitutions 1, (13), (24), (13)(24), which leave ^ = Xi+XiXiXi unaltered. We see that y2=l(i^Ci^-\-'iCi), Ci = 2xi, C2=2^iX2. When a rational function of independent variables Xi, . . . , Xn is unaltered by each substitution of a group H on the re's, but is altered by every substitution not in H, it is said to belong to the group H. We need not specify as in 154 that fl^ is a subgroup of the group G of the equation with the x's as roots, since G is now the total symmetric group. 156. Effect on the Group by an Adjunction to the Domain. Let G be the group of /(x) =0 for a domain i? = i?(^i, . . . , km) containing the coefficients. Let R' = R{^, ki, . . . , km) be the domain composed of the rational functions oi xj/, ki, . . . , km with rational coefficients. This enlarged domain R' is said to be derived from R by adjoining the quantity ^. If the irreducible Galoisian resolvent G{V)=0 for the initial domain R remains irreducible in R\ the group of f{x) = for R' is evidently G. But if it reduces in R', let G'{V) be that factor of G{V) which has its coefficients in R', is irreducible in R', and vanishes for V = Vi. Then if Fi, Fa, . . . , V^ are the roots of G'(F)=0, the groupof/(x)=Ofor/?'is G' = {l,a, . . , *|, a subgroup of G. As a group is included among its subgroups, we have Theorem 1. By an adjunction to the domain, the group of an equation is reduced to a subgroup. 300 GROUP OF AN ALGEBRAIC EQUATION [Ch. XIV Example 1. The group of a;'+x'+x + l = for K{1) is {l, {xiXj)} if i= 1 is the rational root. By adjoining i, we obtain the enlarged domain R({), for which the group is the identity, since now also Xi=i and x=i are in the enlarged domain. The Galoisian resolvent |/i_2r+2 for R(\) has the factors Vi-1 in R{i). Example 2. The group of x< + l = for i?(l) is G,, given by (12). By Ex. 2 of 154, the group for R{i) is G2= fl, (xjXs) (xjX*) } , which is the subgroup of Gt to which i=Xi^=Xs^=Xi^=Xi^ belongs. These examples illustrate also the important Theorem 2. By the adjunction of a rational function ^{xi, . . . , Xn) of the roots with coefficients in the initial domain R, the group of the equation is reduced from G to the subgroup H to which 4' belongs. It is to be shown that H has the characteristic properties A and B ( 149) of the group of the equation for the enlarged domain R'. Any rational function <^ of the roots with coef- ficients in R' equals a rational function ^i of the roots with coefficients in R. First, let <f> be unaltered by all the substitutions of H. By 155, 4>i (which is unaltered by H) is a rational function of ^ with coefficients in R. Hence </)i = is in R'. Hence property A holds for H and R\ Second, let equal a quantity p in R', namely, a rational function p(\f/) with coefficients in R. Then <^i p(^) is a rational function of o^i, . . . , ocn with coefficients in R having the value zero, and hence is unaltered by every substitution of G and, in particular, by the substitutions of H. The latter leave p(^) unaltered and therefore also 0i = <t>. Hence property B holds for H and R\ EXERCISES 1. By the adjunction of V2, the group doi x* + l = for R{t) is reduced to the identity Gi. 2. By the adjunction of an imaginary cube root w of unity, the group Gt of ac' 2 = for R{1) is reduced to the cyclic group C3. Verify that u=Xt/xi belongs to the group C3. By the further adjunction of ^2, the group is reduced to the identity d. 3. Find the group of x*+x^+x^+x+l = for RiVE). CHAPTER XV SUFFICIENT CONDITION THAT AN ALGEBRAIC EQUATION BE SOLVABLE BY RADICALS 157. Solvability by Radicals. An algebraic equation is said to be solvable by radicals if all of its roots can be derived by addition, subtraction, multiplication, division, and extraction of a pth. root (where p has a finite number of positive integral values), these operations being performed a finite number of times upon the coefiicients of the equation or upon quanti- ties obtained from them by those operations. For example, Cardan's formulas (deduced in 158) for the roots of x^+CiXC3=0 are A+B, uA+u^B, w^A+uB, where w is an imaginary cube root of unity and . 3/1 3/1 ^^s c, A=\-cz + Vr, B = \-C3Vr, r= 1 , \2 \2 27 4 A being any definite cube root of \c3-\-Vr, and B being chosen so that AB= C2/3. The radical in w= j+^V 3 is a square root of the num- ber 3 which can be derived from the coefl5cient of 0^ by rational opera- tions. In the above definition we permitted the use of the opera- tion of finding one of the pth. roots of a quantity previously determined, but not the use of the operations of finding all of the pXh roots. The use of the latter operations would imply a knowledge of all of the ;^th roots of unity, whereas we shall prove that the pth. roots of unity are expressible in terms of V 1 and real radicals. The solution of an equation solvable by radicals is often accomplished by the use of a series of auxiliary equations, the roots of any one of which can be found by rational operations 301 302 EQUATIONS SOLVABLE BY RADICALS (Ch. XV and root extractions performed upon its coefficients as well as upon the coefficients and roots of the preceding equations of the series. In order to focus our attention upon a particular equation of a series, and to have a general and hence simpler phraseology, it is convenient to have the following generaliza- tion of the above definition of solvability by radicals. An equation with coefficients in a domain R{ki, . . . , km) shall be said to be solvable by radicals relatively to R if all of its roots can be derived by rational operations and root extrac- tions performed upon ^i, . . . , ^ or upon quantities obtained from them by those operations. For example, *'=2" is evidently solvable relatively to /?(), where is a particular imaginary 13th root of unity. While a quintic equation whose coefficients Ci, . . . , d are independent variables will be shown to be not solvable by radicals, i.e., relatively to R{ci, . . . , Ci), it is solvable relatively to R'=R(xi, Ci, . . . , c^), where Xi is one root of the quintic (since its group for R' is a solvable group of order 24). We have merely shifted the difl&culty to the determination of the new domain R'. The benefit that may be gained by the use of R' is merely one of phraseology. 158. Solution of a Cubic Equation. In a^cix^+C2XC3=0, let Ci, C2, C3 be independent variables. This general cubic equation will be discussed from the group standpoint with the aim of providing a concrete illustration of the general theory which is to follow. Let CO be an imaginary cube root of unity. For the domain R = R{co, ci, C2, cz), the group of the cubic equation is the symmetric group G& on the roots Xi, X2, its ( 153). To the cyclic subgroup C3 belongs the function b = {xi-X2) {X2 - Xz) {Xz - Xi) . By Theorem 2 of 154, 5 is a root of a quadratic equation with coefficients in R. In fact, the discriminant of the cubic equation is 52 = cih2^ -\-\%ciC2Cz - 4^2^ - 4ci%3 - 27C32. 159] RESOLVENT EQUATIONS AND THEIR GROUPS 303 For the domain R' = {R, d), the group of the cubic equation is the cycUc group C3 (156). The substitution (a;iX2a:3) of C3 replaces the functions with coefl5cients in R, by ocf^\{/ and wx, respectively. Thus the substitutions of C3 leave ^ and x^ unaltered. Hence ( 155) the latter are rational functions, with coefl5cients in R, of 5. We have ^4-x3 = a = 2ci3-9ciC2+27c3, ^-x^= -sV^S, ^ = |(a-3V^5). A cube root rj/ of the last quantity is adjoined to R'; the group is thereby reduced to the identity group Gi to which ^ belongs. The roots x are now in the enlarged domain {R', yf). From the expressions for c\, ^, x, we find by multiplications Xi=-\{ci-\-yl^ + x), X2 = \{ci^o^yl^-\-(^x), Xz=\{ci^uyl^-\-(j^x)- Here x = (<^i^ 3c2)/^. In brief, the above solution consists in finding, by means of a quadratic equation, a function ^ which belongs to C3, and then finding, by means of a binomial cubic equation, the function ^ which belongs to Gi. Taking c\ =0, we obtain Cardan's formulas ( 157). 159. Resolvent Equations and their Groups. The auxiUary quadratic and binomial cubic equations employed in the solu- tion * of the general cubic equation are called resolvent equa- tions of the latter. In general, let/(ic)=0 be any given equa- tion with coefiicients in a given domain R, and let ^ be a rational function of its roots with coefficients in R. If ^ belongs to a subgroup H of index v under the group G of f{x) = for R, we have seen ( 154) that ^ is a root of a resolvent equation of degree v with coefficients in R. Suppose that we can solve this resolvent equation relatively to R. By adjoining its root ^ to the domain R, we obtain a domain Ri = (R, rf) for which the group of f{x)=0 is H. If repeated adjunctions lead to a domain R,t for which the group is the identity Gi, * For brevity, we omit the words " by radicals " after " solution " or "solve." 304 EQUATIONS SOLVABLE BY RADICALS [Ch. XV the roots oif{x)=0 will be in R,, (property A of 149). Hence if such a series of resolvent equations can be constructed and solved relatively to their domains, the given equation can be solved relatively to R. Consequently, we shall discuss the question of the solvability of a resolvent equation; to this end we must find its group. By use of (14) in 154, we proved that ^ is one of v conju- gate functions under the group G: and that any substitution 5 of G replaces these by (2) ^ps, h,s, his, , h^s, which are merely the distinct functions (1) rearranged. Hence to any substitution 5 of G on the letters xi, . . . , Xn there corresponds one definite substitution w '=(1 !" !'')-(!" on the V letters (1). Similarly, to t corresponds since we may rearrange at will the letters in the upper line in the two-rowed notation of a substitution. The product <7T replaces yj/g^ by ^^^ and hence corresponds to st. Theorem 1. The substitutions of G correspond to substitu- tions (3) forming a group T. The group r is transitive and isomorphic to G. Example. Let G be the alternating group on the independent vari- ables Xi, . . . , xt. Now \l/={xiX2){x3Xi) belongs formally to the group Gt given by (12) of 150. We have G=Gi +Gi{xiX3Xi) +Gi(xiXat). The indicated substitutions of period 3 replace ^ by 4'! = (xi Xi) (X4 X2), ypi={xi Xi) (xn xs). Since every substitution of Gt leaves also ^2 and ^s luialtered, r=}i, (^1^21^3), i^rj^3^2)\. 1591 RESOLVENT EQUATIONS AND THEIR GROUPS 305 The importance of the group r is due to Theorem 2. r is the group for R of the resolvent equation (4) giy)=iy-H)(y-H) (y-^,)=o. To prove that r has the characteristic properties A and B of the group of (4) for R, note that any rational function p(^i, . . . , \f/,) with coefficients in R equals a rational function r{xi, . . . , Xu) with coefficients in i?: (5) p(^i, . . . , ^f^,)-r{xl, . . . , Xn)=0. Since this difference equals a quantity in R, it is unaltered by any substitution s oi G on xi, . . . , Xn. Since s gives rise to a substitution tr of r on ^i, . . . , ^,, we have (6) Pa(\^i, . . . , 4',)-rs{xi, . . . , Xn)=0. First, let p be unaltered by every substitution of T, so that p = p<r, for every <r in r. Then, by (5) and (6), rs=r for every 5 in G. Hence, by property A for the group G, r is in the domain R. This proves property A for the group T. Next, let p be in R. Then, by (5), r is in R. Hence, by property B for the group G, r=rs for every s in G. Then, by (5) and (6), p = pa for every o- in r. This proves property B for the group r. EXERCISES 1. SiDce r is transitive, (4) is irreducible in R. 2. If G is the symmetric group on iCi, . . . , Xt, the group r on the three y's of Example 2, 150, is the symmetric group of order 6. 3. If G is the symmetric group on Xi, xi, 0C3, and ^ is the alternating function, r is of order 2. The function \l/g belongs to the subgroup g'^Sg of G (9). Hence the v conjugate functions (1) belong to a complete set of conjugate subgroups of G: B, g2-^Hg2, g3-^Hg3, . . . , g,~^Hg,. In case these groups are all identical, 5" is an invariant subgroup of G. In this case, the substitution (3) is the identity if s is in H, since 5 then leaves unaltered each ^^^; while any substitution s of G and any product hs in which hh'vaE corre- 306 EQUATIONS SOLVABLE BY RADICALS [Ch. XV spond to the same substitution (3) of r. Hence, by (14) in 154, we obtain the p distinct substitutions of r by taking those which correspond to s = l, g2, . . , g,- We thus have Theorem 3. If B is an invariant subgroup of G of index V, the group T is a transitive group of order v on v letters and hence is regular. Corollary. // H is an invariant subgroup of G of prime index v, then T is a regular cyclic group of order v. This is illustrated by the above example. Beginning with the group G of the given equation for the given domain R, we can find a series of groups G, H, K, . . . , Gi, terminating with the identity group Gi and such that each is a maximal invariant subgroup of the preceding. If v is the index of H under G, p the index of K under H, etc., the fac- tors of composition of G are v, p, . . . Construct a rational function \l/ of the roots with coef- ficients in R such that 4' belongs to the subgroup H of G. Then )^ is a root of an equation of degree v whose group r for R is simply isomorphic with the simple quotient group G/H. After the adjunction of the root ^ to R, the group of the given equa- tion becomes H for the domain {R, xf/) . Construct a rational function x of the roots with coef- ficients in (R, \f/) such that x belongs to the subgroup K of H. Then x is a root of an equation of degree p whose group for {R, \f) is simply isomorphic with the simple group H/K. After the adjunction of x, the group of the given equation is K. Finally, we adjoin a function belonging to Gi and cbtain a domain containing xi, . . . , Xn. We therefore have Theorem 4. The solution of an equation with the group G for the domain R can be reduced to the solution of a series of equations each with a simple regular group for the domain obtained by adjoining to R a root of each of the earlier equations of the series. If, in particular, G is a solvable group, each auxiliary equation has a regular cyclic group of prime order. 160. Equations with a Regular Cyclic Group. To supple- ment the last theorem we need the result that any equation with a regular cyclic group of prime order p is solvable by radicals. 1601 EQUATIONS WITH A REGULAR CYCLIC GROUP 307 We shall now prove this with the restriction that the domain con- tains an imaginary pih root e of unity. This restriction will be removed in 164 by proving that e can be found by root extractions. Theorem. An equation, whose group G for a domain R containing an imaginary pth root e of unity is a regular cyclic group of prime order p, is solvable by radicals relatively to R. Let Xii, xi, . . . , Xp-i be the roots of the equation and let G be generated by the substitution s = (xoXi . . . Xp-i). Then s replaces the function with coefficients in R, by e'^di. Let Qi = dt^. Then e< is unaltered by 5 and is therefore in R. Thus di is one of the pth roots VOi of a quantity in R. Also the sum of the roots is given by a coefficient of our equation. Thus X<i+Xi-\-X2 + . . . i-Xp-i=c, :ro + eJCi+e2x2 + . . . +e''-'^Xp-i = VQ'i, Xo+e^Xi-]-^X2 + . . . -{-t^''-''xp-i = VQ^, Xo-\-^-^Xi-\-e'^''-'^X2-\- . . . +e^p-i)(p-:rp-i = -^e,-i. Multiply these equations by 1, "', t~^^, . . . , -<^-"-', respectively, add and apply Dividing the resulting equation by p, we get Lagrange's formulas Xt=^\c-\-t-'</e[-\-t-^'<^2+ . . . -{-e-^'-'^'^^e^i] P 0'=o, 1, . . . , p-i). Since the x's are distinct by hypothesis, the O's are not all zero. Thus a certain dt is not zero. Since e may be taken as a new e, we may set 0i5^O. While the first radical may 308 EQUATIONS SOLVABLE BY RADICALS [Ch. XV be chosen as any one of the ^th roots of Gi, the remaining radi- cals are then fully determined. We have where the final factor is in R, being unaltered by s. This proof is illustrated by the final work on the cubic equation (158). 161. Cyclotomic Equations. It remains to treat the equa- tion (7) x''-'^-^x^-^-\- . . . -hx-\-l=0 for the imaginary pth roots of unity, where p is an odd prime. For p = 5, the roots of (7) may be arranged in the order For any prime p, it is shown in the theory of numbers that there exists a primitive root g oi p such that 1, g, g', - , g'-\ . when divided by p, give in some order the remainders 1, 2, 3, ... , p-h Thus the roots of (7) may be written in the order Xi = e, X2 = i', x-i = ^\ . . . , Xp-i = e''^~^. Hence (8) a:2=:J^l^ x-2=X2'', . , Xp-x=xPp-2, xi=o(fp-i, the last relation following from Fermat's theorem that g^~'^ is of the form \-\-kp, where k is an integer. Consider any substitution s of the group G of (7) for R{\) : Xi X2 Xs . ' . Xp-i By (8) and property B of 149, it follows that But by (8). Hence 1621 CYCLOTOMIC EQUATIONS 300 provided the symbol Xp be replaced by xi. Thus h = a+l, c = h-\-l=a-\-2, . . . (mod/>-l), ^^M X2 X^ . . . Xj,-x \ \Xa Xa+l Xa+2 Xa+p-2/ in which Xt+p-i is to be replaced by Xt. Hence s is the power a I of (a:i::2:x:3 . . . Xp-i), so that G is a subgroup of the cyclic group generated by that substitution. After proving in 163 by means of Gauss' lemma that equation (7) is irreducible in R{1), we shall know that G is transitive (150) and h^nce have the important Theorem. // p is an odd prime, the group for the domain of rational numbers of the cyclotomic equation for the imaginary pXh roots of unity is a regular cyclic group of order p \. 162. Gauss' Lemma. If a polynomial f{x) with integral coefficients, that of the highest power of x being unity, is the product of two polynomials with rational coefficients, <t>{x)=xr-\-biC(r-''+ . . . +6, ,A(:^)=x+ciX-'+ . . . +c, then these coefficients are integers. Let the fractions bi, . . . , bm be brought to the least positive common denominator /3o and set &< = ft//3o. Then /3o, . . . , /3m have no common divisor exceeding unity. Sim- ilarly, let Ci = yi/yo, where to, , Tn are integers with no common divisor > 1. Multiplying /=0^ by /3o7o, we get (9) ^oyof{x) = 4>i(x)-Mx), where ^i=/3o^+/3iX~-*4- . . . +/3, ^i=7oa:"+7iX*-*+ +7. We shall assume that j8o7o> 1 and prove that a contradic- tion results from this assimiption. Let /> be a prime divisor of /3o7o. Since p divides each eoefl&cient of the left member of (9), it divides each coefficient of the product <t>i^i. Let ^i be the first coefficient in <i>i{x) which is not divisible by />; let 7t be the first y not divisible by p. The total coefficient of x"+''~*~* in <f>-i}l/i is . . . -f/3+27t-2+/3+i7t-i-|-/3i7t-|-/3-i7jt+i4-/3i-27+2+ 310 EQUATIONS SOLVABLE BY RADICALS [Ch. XV Since A-i, ^-2, . and 7*-i, 7t-2, ... are divisible by p and j8<7t is not, and yet the preceding sum must be divisible by p, we have a contradiction. Hence /3o = 7o = l. 163. Irreducibility of the Cyclotomic Equation. To prove that the function f{x), defined by the left member of (7), is irreducible in the domain of rational numbers, it is sufficient in view of Gauss' lemma to show that f{x) is not the product of two polynomials <l){x) and \J/{x) with integral coefficients, each having unity as the coefficient of the highest power of x. Kronecker's first proof of this fact is essentially as follows: Suppose that such a factorization f{x) = </>(if) ^(a:) is possible. Taking x = l, we get p = (f>{l) \J/{i) . Since /> is a prime, one of these integers, say <^(1), has the value 1. Since the factor *0(x) vanishes for at least one of the roots e, e^, . . . , ^~^ of f{x) = 0, where e is any one of the roots, we have </,() </>(e2) . . . 0(cP-^)=O. In other words, the function P{x) = <t>{x)-4>{x'^) . . . 0(jc^-^) vanishes when x is replaced by any one of the roots of f{x) = 0, and hence has the factor /(x). Thus P{x)=^f{x)-q{x), where q{x) is a polynomial with integral coefficients ( 162, or from the fact that the leading coefficient of the divisor f{x) is unity). Taking x = \, and noting that /(I) =p, we get Since this is impossible, /(ic) is irreducible in R{\). 164. Sufficient Condition for Solvability by Radicals. An equation having a solvable group for the domain defined by the coefficients is solvable by radicals. We shall prove the more general Theorem. An algebraic equation having a solvable group for any domain R containing the coefficients is solvable by radicals relatively to R. 164] SUFFICIENT CONDITION FOR SOLVABILITY 311 This will follow if proved for the case of a regular cyclic group of prime order. Assuming the theorem for this case, let f{x) =0 be an equation whose group for R has the prime factors of composition f, p, . . . As in the proof of Theorem 4 of 159, there is a series of equations w(^) =0, r(x) =0, . . . of prime degrees v, p, . . . , the solution of which is equivalent to the solution of J{x)=Q. The group for R of (^)=0 is a regular cyclic group of prime order v so that this auxiliary equation is solvable by radicals relatively to R. The coef- ficients of r(x) = are in the domain R' = {R, ^) and its group for R' is a regular cycUc group of prime order p; hence it is solvable by radicals relatively to R'. In view of the earlier result, this second auxihary equation is solvable by radicals relatively to R. A repetition of this argument shows that f{x)=Q is solvable by radicals relatively to R. It remains only to prove that an equation C{x)=0 having a regular cyclic group G of prime order p for a domain R is solvable by radicals relatively to R. This is true for p = 2. To proceed by induction, suppose that every equation having a regular cyclic group of prime order <p for any domain D is solvable by radicals relatively to D. As in the proof above, this implies that the equation for the imaginary ^th roots of unity is solvable by radicals (i.e., relatively to the domain of rational numbers). In fact, its group for that domain is a regular cyclic group of order p \ ( 161), each of whose factors of composition is a prime <p. Adjoin to R an imaginary p\h root of unity. The group of C{x) = for {R, e) is either the initial cyclic group G or the identity group. In the latter case, the roots are in {R, e) and can be found from the quantities in R by rational opera- tions and root extractions, since e was shown to be derivable from the rational number by those operations. In the former case, C{x) =0 is solvable ( 160) by radicals relatively to {R, e) and hence, as before, relatively to R. Hence the induction is complete. Corollary. // p is an odd prime, the equation for the p \ imaginary pth. roots of unity is solvable by radicals. 312 EQUATIONS SOLVABLE BY RADICALS [Ch. XV The theorem implies that any cubic equation is solvable by radicals, since its group for any domain containing the coeffi- cients is solvable. 165. Solution of a Quartic Equation. Let the coefficients in 0(!^-\-ax^-\-bx^-\-cx+d = O be independent variables, so that we have the general quartic equation. Its group for the domain R = R{o3, a, b, c, d), where w2 -|- CO -|- 1 =0, is the symmetric group G24 on the roots ici, . . . , x^ ( 153). It is a solvable group, having the factors of com- position 2, 3, 2, 2. In view of the last theorem, the equation is solvable by radicals. We shall give a solution which will illustrate the developments of the general theory as presented in the next chapter. In fact, we shall employ only binomial resolvents. To the invariant subgroup G12 composed of the even substi- tutions belongs the function 8 = {XiX2){xi-X3)iXlX4:){X2Xs){X2X4){X3X4), whose square equals * the discriminant of the quartic: 62 = 256(/3-27/2), ,_ ,_a , _6^ T_bd_ c^ _a^d abc b^ 4 12' ~~Q I6~T6 l8~216* The group of the quartic equation is G12 for {R, d). Employ the notations of Example 2, 150. To the invariant subgroup G4 of G12 belongs 0=yi + w3'2 + w2y3, whose conjugates under G12 are co<^ and w^^. Thus </> is a root of the resolvent cubic 2^ <l>^ = 0. To find <f>^, set \l/=yi-\-i>Py2-\-o}y3. Then ^3_^3 = 3('^_^,2)(-yj_y2)(>'2->'3)(3'3-yi)=-3(co-a>2)6, ^-\-<t>^ = 2{yi^-\-y2^-^y3^)-\-12yiy2y3-\-3(o>-\-or')v, * Dickson, Elementary Theory of Equations, p. 42, Ex. 7. 165] SOLUTION OF A QUARTIC EQUATION 313 where V =yi^y2 -\-yiy2^+yi^y3-hyiy3^ +>'2^3'3 +y2>'3^. But C^yiY =^v-\-Qyiy2y3-\-^yi^, Thus ^ + 03 = 2(2>'i)'-9(2^O i^yy^) +27yi3;2>'3 = 263 - Qbiac - M) +27(c2 4-a2(f - ^bd) = - 432/, as seen from the ^'-cubic in the example referred to. Hence After the adjunction of <l>, the group is G4. Since yi, y2 and y3 are unaltered by G4, they are in the new domain {R, 8, 4>). To find them, note that <t>^ = ^yi'+(<^+^')^yiy2={^yiy-^^yiy2 = i2l, 12/ yi+y2-\-y3 = b, yi-\-o)y2-]-o}^y3 = <t>, yi-]-oPy2-\-o}y3=-. Hence The square of h=xi-\-X2X3X4: is (^ a;i)^ 4^a;ia;2+4yi=a^ 46+4yi- Adjoining a root of h^ = a^-^b+4yi, we obtain a domain (R, 8, 0, ^1) for which the group of the quartic equation is G2 = {1, ixiX2){x3X4)\- Adjoining also a root t2 = Xi+X3X2X4 of t22 = a^-4b-\-4y2, 314 EQUATIONS SOLVABLE BY RADICALS [Ch. XV we have a domain for which the group is the identity Gi, to which therefore xi, . . . , xa belong. To find them, note that h^ = a^ 4b+iys, t3=Xi-\-Xi-X2-X3, as shown by setting y = b a^/4: in the cubic function in the example referred to, which equals n(y yj). To determine the sign of the square root, take a;i = l, Xg = 0{g>l); thus ht2h= 8c+4a& a^. Subject to this condition, the roots are given by Axi = -a+h-\-t2+h, ^2=-a-\-h-t2-h, 4iC3= a h-\-t2 h, ^X4:= a h t2-\-h. The above solution can be modified indefinitely, since there are infinitely many functions belonging to a given sub- group of the equation. Moreover, we might employ other subgroups of order 2 of G4 instead of G2. EXERCISES 1. After reducing the group to G2, adjoin j=\/^ and find the quad- ratic equation for Vxi X2 + ixi ixi'. V^={Xl-0Ciy-{X3-X^y+2i{Xi-Xi){xz-X^) -8c o 2 h 3 , 4ofe-8c-o' 2 ^/ 12/\' For W=Xix-iix3-\-iXi, we have VW={x,-Xiy-\-{x3-x,y=TiXi''-2y,=a^-'lh-2yi, X,, Xi=\{-a+h{V+W)], X,, X3=-H-o-/. ^-'(F-Pr)]. The theoretical interest of this solution is that it leads to a 24- valued function V by means of a chain of binominal resolvents. 2. After reducing the group to G2, we may find Xi and x^ by solving a quadratic equation with coefficients in (R, 5, 4>, h) : x^+h{a-ii)x+hi-ihayi-c)/ii = 0. In terms of Xi, Xt and the quantities known, we may find x, Xti ytyi Xi-\-Xi = Xi+X2 ti, X3 Xi = ' X\ X% CHAPTER XVI NECESSARY CONDITION THAT AN ALGEBRAIC EQUATION BE SOLVABLE BY RADICALS 166. Galois' Criterion. An algebraic equation is solvable by radicals if and only if its group for the domain defined by the coeffixients is a solvable group. It is occasionally useful to employ the generalization: An equation is solvable by radicals relatively to a domain R containing the coefficients if and only if its group for R is solvable. That the solvability of the group is a sufficient condition for the solvability of the equation was proved in 164. We shall now prove that it is a necessary condition. By hypoth- esis the roots xi, . . . , ocn oi the equation can be derived by rational operations and root extractions from quantities in the domain R = R{ki, . . . , km) or from quantities obtained from them by those operations. The index of each root ex- traction may be assumed to be prime, since z"' = a is equivalent to the pair of equations z^=w, lif =a. li ^, r], . . . , \j/ denote the radicals which enter the expressions for a:i, . . . , Xn, the solution may be exhibited by a series of binomial equations |^=L(^l, . . . , km), r = Mi^, kl, . . . , km), . . . , V=S{. . . , V, ^, kl, . . . , k^) of prime degrees \ n, . . . , &, together with equations which express xi, . . . , Xn rationally in terms of ^, . . . , ^, ^i , . . , Here L, . . . , S are rational functions with rational coefficients of their arguments. Consider, therefore, a binomial equation of (2) x''-A=0, 315 prime degree 316 EQUATIONS SOLVABLE BY RADICALS [Ch. XVI where A is in the domain R. Let be an imaginary p\h root of unity. If one root r of (2) belongs to the domain R' = {R, c), all of the remaining roots er, e^r, . . . , ^"V belong to R' and the group of (2) for R' is the identity. In the contrary case, A is not the />th power of a quantity in R', and (2) is irre- ducible m. R' ( 143). The notation of the roots can be chosen so that .By an argument like that of 161 with p 1 replaced by p, we see that the group of (2) for R' is a subgroup of the cyclic group generated by (jcia!;2^3 . . . Xp). Since (2) is now irre- ducible in R', its group is transitive and hence of order = p. For a domain containing A and an imaginary pth root of unity, the group of the binomial equation (2) of prime degree p is the identity group if any root is in the domain, but is a regular cyclic group of order p if no root is in the domain. For examples, see 150. Any one of the binomial equations (1) of prime degree p is equivalent to a series of equations each having a regular cyclic group of prime order for a certain domain. We include in the series the equations having regular cyclic groups of prime orders dividing p l, which together serve to determine an imaginary pth root e of unity ( 161, Theorem 4 of 159). After the adjunction of e, the group of the binomial equation was just proved to be either the identity group or a regular cychc group of order p. In the former case, the desired series of equations is the one previously defined; in the latter case, that series together with the given binomial equation. Thus the set of binomial equations (1) is equivalent to a series of equations of prime degrees, each having a regular cyclic group for the domain symbolized in the same line : <^(y; ki, . . . , k)=0, R = R{ki, . . . , hi)] rp{z; y, ku . . . )=0, (y, R); e{w; . . . z, y, ki, . . . )=0, (. . . 2, y, R). 1671 THEOREMS OF GALOIS, JORDAN AND HOLDER 317 Each of these equations is solvable by radicals relatively to the corresponding domain ( 164). Adjoin one root y of <> = to R\ the group is now a subgroup R of the initial group G, including the possibiUty that E.=G (156). Solve ^ = and adjoin one root z to the domain {y^ R); the group is now a subgroup of H. Proceeding in this manner, we finally reach the domain {w, . . . , z, y, R) containing each root Xt of the proposed equation, whose group is therefore now the identity group Gi. The theorem of Galois states that, by each of these adjunc- tions, the group of the proposed equation is either not reduced at all or else is reduced to an invariant subgroup of prime index. This theorem will be derived as a corollary in the next section from a theorem of which other important apphcations will be made later on. Hence the distinct groups G, H, . . . , Gi, obtained by the successive adjunctions, form a series of composition of G with only prime factors of composition. Thus G is a solvable group. We therefore have Galois' criterion for the solvability of an equation. For w>4, the factors of composition of the symmetric group on n letters (17) are 2 and ^nl, the latter of which is not prime. By 153 the group of the general equation of degree n, i.e., one whose coefficients ci, . . . , Cn are independent complex variables, is the symmetric group when the domain is that defined by ci, . . . , c and a finite number of constants. We therefore have the Theorem. The general equation of degree n>^is not solvable by radicals * Moreover, its roots cannot be found by rational operations and root extractions perfortned upon the coefficients and any constants, finite in number, or upon quantities obtained from them by those operations. 167. Theorems of Galois, Jordan and Holder. Of prime importance is Jordan's Theorem.! Let the group Gi for a domain R of * Ruffini, Teoria generde ddle equazioni. . . , Bologna, 1799. N. H. Abd, (Euvres, vol. 1, 1881, pp. 66-94. t Jordan, TraiU des substitutions, p. 268-9. 318 EQUATIONS SOLVABLE BY RADICALS (Ch. XVI an algebraic equation Fi{x)=0 be reduced to G'l by the adjunc- tion of all of the roots of a second equation Foix) =0, and let the group G2 for R of the second equation be reduced to G'2 by the adjunction of all of the roots of tlie first equation. Tlien G'\ and G'2 are invariant subgroups of G\ and G2, respectively, of equal indices, and * the quotient-groups Gi/G'i and G2/C2 are simply isomorphic. By 154 there exists a rational function ^1 with coefficients in R of the roots ^1, . . , ^i, of the first equation, such that ^1 belongs to the subgroup G\ of Gi. Since the adjunction of the roots rji, . . . , ?; of the second equation reduces Gi to G'l, property A of G'l ( 149) shows that ^1 Ues in the enlarged domain : (3) ^l(h, . , $) = 01(171, . . . , Vm), where <t>i is a rational function with coefficients in R. Let ^i, \l/2, . . , ^t denote all of the numerically distinct values which \f/i can take under the substitutions (on ^1, . . . , ^n) of Gi. Then G'l is of index k under Gi ( 154, Theorem 2). Let 01, . . . , 0j denote all of the numerically distinct values which 01 can take under the substitutions (on 771, ... , 7?^) of G2. The k quantities rp are the roots of an equation irre- ducible in R; likewise for the / quantities 0. Since these two irreducible equations have a common root i/'i = 0i, they are identical ( 144). Hence the ^'s coincide in some order with the 0's; in particular k=l. If Si is a substitution of Gi which replaces ^1 by ^1, then Si transforms the group G'l of \}/i into the group of ^i of the same order as G'l. Since xf/i equals a 0, it is in the domain R' = {R, 771, . . . , Tjm) and hence is unaltered by the substi- tutions of the group G'l of Fi{x) = for that domain R' ( 149, property B). Hence the group of ^< contains all of the sub- stitutions of G'l and, being of the same order, is identical with G'l. Thus G'l is invariant in G\. The group for R of the irreducible equation satisfied by ypi is therefore the quotient group Gi/G'i ( 159). * This supplement and the proof here employed arc due to HOlder, Mathe- matische Annalen, vol. 34, (1889), p. 47. 1671 THEOREMS OF GALOIS, JORDAN AND HOLDER 319 Let H2 be the subgroup of G2 to which 01(171, . . . , ij,) belongs. Since 4>i is a root of an equation of degree l = k irre- ducible in R, the group H2 is of index k under G2. By the adjunction of </>i, i.e., of ^1 by (3), the group G2 of F2{x)=Q for R is reduced to ^2 ( 156, Theorem 2). If not merely ^1(^1, . , l), but all of the $'s themselves be adjoined, the group G2 reduces perhaps further to a subgroup of H2. Hence G'2 is contained in E2. We thus have the preliminary result: If the group of Fi(a:)=0 reduces to a subgroup of index k on adjoining all of the roots of F2{x)=Q, then the group of F2{x) =0 reduces to a subgroup of index ^1, ^1 ^ ^, on adjoining all of the roots of Fi(x)=0. Interchanging Fi and F2 in the preceding statement, we obtain the result: If the group of F2{x) =0 reduces to a subgroup of index ki on adjoining all the roots of Fi{x) =0, then the group of Fi{x)=Q reduces to a subgroup of index ^2, ^2^^i, on ad- joining all the roots of F2{x) =0. Since the hypothesis for the second statement is identical with the conclusion for the first statement, it follows that k2 = k, ki = k, ^2 = ^1, so that ki = k. Hence the group G'2 of the theorem is identical with the group H2 of all of the substitutions in G2 which leave <f>i unaltered. For the same reason that G'l is invariant in Gi, it now follows that G'2 is invariant in G2. The equation irre- ducible in R and satisfied by 0i has as its group the quotient- group G2/G'2. Since the two irreducible equations in R satisfied by <f>i and \f/i, respectively, were shown to be identical, their groups Gi/G'i and G2/G'2 differ only in the notations employed for the letters on which they operate, and hence are simply isomorphic. We shall derive as a corollary Galois' Theorem. By the adjunction of any one root of an equation 7^2 (x) =0 whose group for R is a regular cyclic group of prime order p, the group for R of the equation Fi (x) = either is not reduced at all or else is reduced to an invariant subgroup oj index p. 320 EQUATIONS SOLVABLE BY RADICALS [Ch.XVI In fact, by adjoining one root xi of F2(x)=0, we adjoin all of its roots, since each is a rational function of xi with coef- ficients in R ( 155). For, the identity is the only substitution of the group for R of F2{x)=0 which leaves xi numerically unaltered. We shall state certain results not presupposed in what follows. A brief argument (cf. Dickson's Theory of Algebraic Equations, 1903, p. 83) now leads to Abel's theorem: The roots of an equation solvable by radicals can be given such a form that each of the radicals occurring in the expres- sions for the roots are expressible rationally in terms of the roots of the equation and certain roots of unity. This was proved by Abel by a long algebraic discussion without the aid of groups and employed in his proof of the impossibility * of solving by radicals the general equation of degree For a domain R an irreducible equation of prime degree whose roots are all rational functions of two of the roots with coefficients in R is called a Galoisian equation. Galois proved that it is solvable by radicals and that every irreducible equation of prime degree which is solvable by radicals is a Galoisian equation. For a detailed exposition with illustrative exam- ples, see Dickson's Theory of Algebraic Equations, 1903, pp. 87-93. A cubic equation having three real roots cannot f be solved by real radicals (the " irreducible case "). * Cited in 166. Cf. Scrret, Cours d'Algibre supirieure, ed. 4, vol. 2, pp. 497-517. t H. Weber, Algebra, ed. 2, 1898, vol. 1, 657; Kleines Lehrbuch der Algebra, 1912, p. 381. CHAPTER XVII CONSTRUCTIONS WITH RULER AND COMPASSES 168. Some Celebrated Problems of Greek Origin. In the Dalian problem of the duplication of a cube, we are given the length s of an edge of a cube and seek to construct by ruler and compasses the edge x oi a. cube whose volume is double that of the first cube. For this problem, as well as for the problem of the trisection of an arbitrary angle, and for the problem of the construction of a regular polygon of 7 or 9 sides, the ancients sought in vain for constructions by ruler and compasses. The impossibility of these constructions was proved only in recent times. To the analytic methods employed in the proof of this impossibility is due also the discovery of new constructions, such as that for the regular polygon of 17 sides, the constructibiUty of which was not sus- pected during the twenty centuries from Euclid to Gauss. 169. Analytic Criterion for ConstructibiUty by Ruler and Compasses. The first step in our treatment of the problems mentioned in 168 is their analytic formulation. In the Delian problem, we are led at once to the equation j^ = 2s^. Next, if angle 120 could be trisected or if a regular polygon of 9 sides could be constructed by ruler and compasses, angle 40 could be constructed and hence cos 40. In the identity cos SA=4: cos^ AZ cos A , take ^ = 40 . Since cos 1 20 = - , we get 4 cos3 40 - 3 cos 40 -|-^ = 0. Multiply by 2 and set a: = 2 cos 40; thus (1) r-3a;+l=0. 321 322 CONSTRUCTIONS BY RULER AND COMPASSES [Ch. XVII In this problem and- the Delian problem, we are given the coefficients of a cubic equation and ask whether or not a line whose length is a root x can be constructed by ruler and com- passes. We shall first prove that an affirmative or negative answer is to be given according as x can or cannot be derived from the coefficients by rational operations and extractions of real square roots. For any proposed construction we are concerned with certain numbers, some expressing lengths, areas, etc., others being the coordinates of points, and still others being the coef- ficients of equations of straight lines or circles referred to rect- angular axes. We shall establish the Criterion. A proposed constrtiction by ruler and compasses is possible if and only if the numbers which define analytically the desired geometrical elements can be derived from those defining the given elements by rational operations and extractions of real square roots performed a finite number of times. First, let the construction be possible. The straight lines and circles drawn in making the construction can be located by means of points either initially given or obtained as the intersections of straight lines and circles. The coordinates of the intersection of two intersecting lines are evidently rational functions of the coefficients of the equations of the lines. If the straight line y = mx-\-b intersects the circle the coordinates of the points of intersection are found by elim- inating y, solving the resulting quadratic for x, and inserting the roots x into y = mx-\-b. Hence the coordinates are found from m, b, p, q, r by rational operations and the extraction of a single real square root. Finally, two intersecting circles cross at the intersections of one of them with their conunon chord, so that this case reduces to the preceding. That the criterion gives also a sufficient condition for con- structibility is shown by the facts that the sum or difference of two segments of straight lines can be found by use of com- passes, that p = ab can be found by constructing p in 1 i a = b : p $ 1721 REGULAR POLYGONS 323 by use of parallels, and similarly q = a/h in 1 : b = q : a. Finally, if n is a positive number, V can be constructed by the use of a semicircle of diameter l+ and a perpendicular at the point separating the segments of lengths 1, n. 170. Trisection of an Angle. To prove that it is impossible to trisect an arbitrary angle by ruler and compasses, it suffices to prove that angle 120 cannot be trisected. We saw that 2 cos 40 is a root of (1). In the domain of rational numbers, Eq. (1) is irreducible ( 144, Ex. 3) and has the discriminant 81 ; hence its group is of order 3. By the adjunction of a square root, the group is either not reduced at all or is reduced to a invariant subgroup of index 2 ( 167, Galois' Theorem). Hence no such reduction is possible in the present case. If therefore the cubic had a constructible root, its adjunction would cause no reduction of the group, whereas the adjunction of any root reduces the group to the identity. 171. Duplication of a Cube. If an edge of the cube be taken as the unit of length, the edge of the desired cube is a root of For the domain of rational numbers this irreducible equation has as its group the symmetric group Ge. The adjunction of any root reduces it to a group of index 3. Hence no root can be found by extractions of square roots. 172. Regular Polygons. The construction of a regular polygon of n sides by ruler and compasses is equivalent to that of angle 2ir/n and hence of a line of length cos 2ir/w. The irreducible equation with rational coefficients satisfied by the latter number is much more difficult to form and treat than that with the root (2) f = cos h* sin , n n where i = Vl. In view De Moivre's theorem, r is an nth root of unity. Moreover, 1 2X . . 27r ,1 o^^e^T - = cos tsin . r-\ = 2 cos . r n n r n 324 CONSTRUCTIONS BY RULER AND COMPASSES [Ch. Xvll Hence if r can be expressed in terms of i and real square roots, cos 2'K/n can be expressed in terms of real square roots. The converse is seen to be true by an inspection of (2), since the sine can be found from the cosine by a real square root. Hence a regular -gon can be constructed by ruler and compasses if and only if the th root (2) of unity can be found by the ex- traction of square roots, all except the last one of which is real. If n is an odd prime ^, r is a root of an equation of degree p \ irreducible in the domain R of all rational numbers and having as its group for R a regular cyclic group C of order /> 1 ( 161-3). The adjunction of any root reduces C to the identity. If a regular p-gon can be constructed, the ad- junction of the root r is equivalent to that of several square roots, the adjunction of each of which causes either no reduction in the group or a reduction to a subgroup of index 2. Hence a regular ^-gon can be constructed by ruler and compasses if and only if p \ is a power 2* of 2. But if /j =Jq, where / is odd, then 2''-fl has the factor 2*' + l- Hence a prime of the form 2''4-l is of the form (3) 22' -fl. For t = 0, 1, 2, 3, 4, the corresponding numbers are 3, 5, 17, 257, 65537, and are all primes. But for / = 5, 6, 7, 8, 9, 11, 12, etc., the number is known to be not prime. Next, let n = ab, where a and h are relatively prime integers > 1. If a regular a-gon and a regular 6-gon can be constructed by ruler and compasses, the same is true of a regular w-gon. For, multiples of the angles 27r/a and 2Tr/h can then be con- structed and hence also the sum of these multiples. Since there exist integers c and d such that ca-\-dh = \, the angle , 27r 27r 27r. ,, s 27r d \-c- =-{dh+ca) = , a ab ab and therefore also the a6-gon, can be constructed. Conversely, from the latter we obtain a regular a-gon by using the 1st, (6-fl)th, (26 + l)th, . . . , [(a-l)& + l]th vertices. Hence if n = f(l . . . , where p, q, . . . are distinct primes, a regular 172] REGULAR POLYGONS 326 n-gon can be constructed if and only if a regular p^-gon, (f-gon, . . , can be constructed. A 2*-gon can be constructed by repeated bisections of 180. It therefore remains only to discuss the regular p^-gon, where p is an odd prime. By De Moivre's theorem, p = cos .+z sm -, P' f is a root of x" =1, but not oi x^ =1, and hence is a root of (4) ^!!l^:^(p-i)+a:'(i'-2)+ . . . +^_^1=0 (/ = />*-*). Since p^, p^", . . . , p'" give the t roots of a:' = l, the remaining tp t powers of p, with positive exponents less than tp and not divisible by p, are roots of (4) and give all of the roots of (4). They are called the primitive />*th roots of unity. For p*=9, the six primitive ninth roots of unity are p, p*, p*, p*, p% p' and are the roots of x^+x^+l = 0. The proof that (4) is irreducible in the domain R of all rational numbers differs from that in 163 for the special case 5 = 1 only in the detail of having, instead of c, c^, . . . , e**"^ in the former case, the roots p, p", p*, . , . , p' of (4), where I, a, b, . . . , / denote the positive integers less than />* and not divisible by p, and p is an arbitrary primitive p^th root of unity. As shown in the theory of numbers, there exists a primitive root g of f, where p is an odd prime, i.e., an integer g such that 1, g, g^ . . . , ^-' ik=p'-p'-'), when divided by p", give as remainders in some order the posi- tive integers less than ^ and not divisible by p. Thus the roots of (4) are p, p*, p'*, . . . , pT . In the former example p' = 9, we may take g = 2. Then the preceding roots are p, p*, p*, p', p', p', respectively. Since each root of (4) can therefore be expressed as the ^th power of the preceding root, we readily find as in 161 326 CONSTRUCTIONS BY RULER AND COMPASSES [Ch. XVII that the group of (4) for 2? is a regular cyclic group of order k. li s>l, k is not a power of 2, and by the usual argument the regular />'-gon cannot be constructed by ruler and compasses. Combining our results, we have the Theorem. A regular polygon of n sides can he constructed by ruler and compasses if and only if n = '^p\p2 , where pi, P2, . . . are distinct primes of the form (3). Since therefore a regular 9-gon cannot be constructed, we have a new proof that angle 120** cannot be trisected by ruler and compasses. Gauss * was the first to prove that a regular p-gon can be constructed if /> is a prime of the form (3) ; he stated,t but apparently did not publish a proof of, the remaining part of the above theorem. For the elegant method invented by Gauss for finding the series of quadratic equations leading to a 17th root of unity and the actual geometrical construction of a regular 17-gon, as well as for a longer proof of the above theorem without the aid of group theory, the reader may con- sult the monograph by Dickson, { where references to other books are given. * Disqidsitiones ArithmetioB, 1801, Art. 335-366 [=Werke, 1]; German translation by Maser, 1889, pp. 397-448, 630-6.52. t Gauss-Maser, p. 447. } Monographs on Modern Mathematics, edited by J. W. A. Young, New York, 1911. A brief, but more elementary, treatment is given in Dickson's Elementary Theory of Equations, 1914, pp. 84-92. A still more elementary discussion is that by Dickson, Amer. Math. Monthly, vol. 21 (1914), 259-262. CHAPTER XVIII THE INFLEXION POINTS OF A PLANE CUBIC CURVE 173. Homogeneous Coordinates of Points in a Plane. Let aiX-\-biy-\-Ci=0 (* = 1, 2, 3) be any three linear equations such that ai bi c\ A= a2 62 C2 ?^0. a3 63 C3 Interpret x and y as the Cartesian coordinates of a point referred to rectangular axes. Then the three equations represent three straight lines Lt forming a triangle. Choose the sign before the radical so that _ (hx-\-hiy-\-Ci is positive for a point {x, y) inside the triangle, and hence is the length of the perpendicular from that point to L^. The homogeneous coordinates of a point {x, y) are three numbers x\, X2, X3 such that pXi=kiP\, pX2 = k2p2, pxs hpsj where ki, ^2, ks are constants, the same for all points, while p is an arbitrary factor of proportionality. Thus only the ratios of xi, X2, X3 are defined. The coefficients of the linear function ktpi, which are proportional to Ot, 6<, c<, will hence- forth be denoted by those same letters. Then (1) pXi = aiX-hbty-\-Cu Aj^^O (t = l, 2, 3). Solving these equations by determinants, we get Ax = p^AiXi, Ay = p^BtXi A = p^C,a;<, 327 328 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIIl where Ai is the cofactor of a in A, 3% that of bt, and d that of Ci. Hence (2) x^^ ^2^ 2 Q^ 2 CiOCi Thus any equation /(a;, y)=0 can be expressed as a homoge- neous equation <i>{x\, X2, a;3)=0 of the same total degree, and conversely. In particular, any straight line is represented by an equation of the first degree in xi, X2, Xs, and conversely. For example, a:i = represents a side of the triangle of reference. Let yi, y2, y3 be the homogeneous coordinates of the same point (x, y) referred to a new triangle of reference having the sides L'i. As before, (lO pyi = a\x+b\y+c\ (f = l, 2, 3), where the right member equated to zero represents L'i. Solv- ing equations (1') as we did (1), we obtain x and y as Unear frac- tional functions of yi, y2, ys- Inserting these values into (1), we get formulas like (3) Xi-=Ciiyi-{-Ci2y2+Ci3y.i, \cij\^0 (z = l, 2, 3). Thus a change of triangle of reference gives rise to a linear transformation of the homogeneous coordinates. Let/(xi, X2, xs) be a homogeneous rational integral function of the wth degree. Under the transformation (3), let it become <l>(yi) 3'2, y^)- Then </> = represents the same curve as/=0, but referred to the new triangle of reference. Let t = kxi<'X2^X3'' {a-\-b-\-c = n) be any term of/. Then Xi = at, X2 = bt, X3 =ct. dxi dX2 dX3 Their sum is nt. Hence we have Euler's theorem: (4) x,^-\-x^-i-x;-^ = nf. dxi dX2 dxs 174] HESSIAN CURVE 329 If xi, X2, X3 is a set of solutions, not all zero, of (5) ^ = 0, -^ = 0, -^ = 0, ^ ^ dXl dX2 3X3 ' and hence by (4) of /=0, the point {xi, xi, x-^ is called a singular point of the curve /=0. At this point. dyj ^ dxi dyj C/ = l,2,3). Hence the definition of a singular point is independent of the special triangle of reference chosen. It is readily proved, but not presupposed in what follows, that two or more branches of the curve pass through any singular point, which is there- fore called a double or multiple point. 174. Hessian Curve. The Hessian of / is h = dxi^ 8x28x1 ay d'f dj dxi 8X2 8X2^ 8X1 8X3 8X28X3 8x38x1 8x38x2 8x3^ Let transformation (3), of determinant A, replace / by <t>(yi, y2, yz)' The product hA is a determinant of the third order, in which the element in the ^*th row and yth column is the sum of the products of the elements of the ith row of // by the corre- sponding elements of the^'th column of A, and hence is 8Y 8x<8xi cir 9'/ 8x,8x2 C2r m 8x<8x3 Czj. The latter is the partial derivative with respect to x< of 9/ 8xi 8/ 8X2 8/ 8X3 ^80 8xi dyt 8x2 dyj 8x3 8y; 8y/ Let A' be the determinant obtained from A by interchanging its rows and columns. By the same rule of multiplication, = Hessian H of 0. r. J-l, 2.3 330 INFLEXION POINTS OF A CUBIC CURVE (Ch. XVIII the element in the rth- row and jth. column of the determinant equal to A' AA is ^'dxAdyJ ^'dx2\dyj/ "'dx-sKdyJ dyXdyJ' since c, is the partial derivative of Xi with respect to yr. Hence I dyrdyj In words, A^h becomes // under the transformation (3), so that H = represents the same curve as h = 0, but referred to the new triangle of reference. Hence there is associated with any curve /=0 a definite Hessian curve h = independent of the choice of the triangle of reference. 175. Points of Inflexion of a Cubic Curve. Let/(a;i, X2, xs) be of the third degree. Choose a triangle of reference having the vertex P = (0, 0, 1) at a point on the curve /=0, not a singu- lar point. Then there is no term involving xs^, and the coef- ficients of the terms rxiXs^ and SX2X3^ are not both zero, since otherwise the derivatives (5) would all vanish at P. Hence we may take rxi-^sx2 as a side of a new triangle of reference with the same vertex P and obtain X3^xi-\-X3{axi^-\-bxiX2-\-cx2^)-\-<t>(xi, 3:2) =0 as the new equation of our curve. Replacing xs by X3-{axi-j-bx2)/2, we get Fi=xs^xi-{-ex3X2^-^C{xi, X2). Denote the second derivative of the cubic function C with respect to oci and Xj by Cy. Then the Hessian of Fi is Cii C12 2x3 Hi = C21 C22-\-2ex3 2ex2 = -8e:r33-F 2x3 2eji;2 2^i Hence P = (0, 0, 1) is on Fi =0 if and only if e = 0. If d is the coefficient of X2^ in C, then xi ~0 meets Pi = in the points for which X2^{ex3-^dx2)=0, and these three points 175] INFLEXION POINTS AND TRIANGLES 331 coincide (at P) if and only if e = 0. In that case P is called a point of inflexion of Fi = and Xi = {) the inflexion tangent toFi=Oati'. Thus P is a point of inflexion of Fi =0 if and only if it is on ni=0. Hence, by 174, each intersection of a cubic curve /=0 without a singular point with its Hessian curve h = is a point of inflexion off=0, and conversely. There is certainly at least one intersection. For, by elim- inating X3 between /=0 and h = 0, we get a homogeneous equation in xi and X2, having therefore at least one set of solu- tions x'l, x'2. Then, for X\=x'i, iC2=ic'2, the equations /=0, /f = have at least one common root x=x'z. Thus {x'\, x'2, x'i) is an intersection and therefore a point of inflexion of /=0. Taking this point as a vertex (0, 0, 1) of a triangle of reference and proceeding as before, we get F of type Fi with e = 0. If the coefficient d of X2^ in F is zero, F has the factor x\. But, if F=xiQ, the derivatives dxi dxi dX2 dX2 dxs dX3 all vanish at a point of intersection of 0:1 = 0, Q=0, whereas F = has no singular point. Hence d9^0. Taking d^X2 as a new X2, and then adding a suitable multiple of Xi to X2 to delete the term with X2^^i, we get F=X3^Xi+C,' C=X2^-^dbx2Xi^-{-axi^, H = 2xi Cii C12 C21 C22 -^X3^C22 = 72Xi bx2-{-axi bx\ bX\ X2 -24X32X2. Eliminating x^^ between F = 0, ^ = 0, we gt xi C X2 ^x\{bx2^-\-axiX2 l^x\^) = X2*+Qbx2^Xi^-\-^ax2Xi^ - 362x1* = 0. If xi=0, then X2 = and the intersection is (0, 0, 1). For the remaining intersections, we miay set xi = 1 ; then each root of (6) f*-|-6&r2+4ar-362 = 332 INFLEXION POINTS OF A CUBIC CURVE (Ch. XVIII leads to the inflexion points (1, r, j), where -52 = C = r3+36r+a. If 5 = 0, (6) would have a double root and a^-\-4i^ = 0. But the partial derivatives of F all vanish at (1, X2, 0) if r:2^+6 = 0, 2bx2+a = 0, and hence if b = 0, a[;2 = 0, or if 67^0, X2= a/{2h), whereas F has no singular point. Hence (7) a2+4JV0 and (6) has four distinct roots r for each of which S9^0. Thus there are exactly nine distinct points of inflexion. The two points (1, r, s) with a fixed r are collinear with P = (0, 0, 1), being on X2 = rxi. For the remaining roots p of (6), we have p8 +rp2 + (r2 +66) p+r3 4.6 J;. +4^ = 0. The product of this by r can be written in- the form r(p3+36p+a) + (rp+^)2 = 0, k = h{r'-\-Sb). Hence the quadratic factor in (8) ^H-\-rF = (rxi-X2){xs^-krx2+kxiY\ vanishes at (1, p, 0-), where -<r2 = p3+36p+a. Thus the nine points of inflexion lie by threes upon the three straight lines given by (8), which are said to form an inflexion triangle. There are four inflexion triangles, one for each root r of (6). The roots of (6) are the only values of r for which H-\-2irF has a linear factor /. In fact, / = meets F = in three points on n = which are therefore points of inflexion. Thus / has two companion linear functions such that llil2 = is one of the four inflexion triangles. Hence llil2 = H-)r2ApF, where p is one of the roots of (6). By hypothesis, lQ = n-{-2irF. If 176] GROUP FOR THE INFLEXION POINTS 333 rj^p, we see by subtraction that F has the factors / and Qlih and hence has a singular point, contrary to assumption. Corresponding results follow at once for the general cubic curve /=0. We saw that/ can be reduced to F by a linear transformation of a certain determinant 5. But F is replaced by a Uke form by the transformation which multiplies xi, X2, X3 by 8~^, 1, 8, respectively, and thus has the determinant 8~^. The product of the two transformations is of determi- nant unity and replaces/ by a form F. Hence ( 1 74) , it replaces the Hessian h oi f by the Hessian H of F. Thus for each root r of (6) , in which a and b are certain functions of the coefficients of/, A+24r/=0 represents an inflexion triangle of/. Furthermore, a and b are rational functions of the coefficients of/. For, there are exactly four values of r for which (f>=h-\-2Arf has a linear factor xi mxonxs. Replacing xi by mx2-\-nx3 in (f), we obtain a cubic function of X2 and x-s whose coefficients must vanish. Eliminating m and n, we obtain two equations in which r and the coefficients of / enter rationally and inte- grally. The greatest common divisor of their left members must be a function of r whose coefficients are rational in those of/. The latter is therefore true ( 145, first foot-note) of the quartic equation * for r with no multiple root. 176. Group G of the Equation X for the Abscissas of the Points of Inflexion. Let R be the domain defined by the co- efficients of the equation /=0 of a cubic curve without singular points. We employ a new triangle of reference whose side ii;i = does not contain a point of inflexion. This can be ac- complished by a Unear transformation on Xi, X2, xs with coef- ficients in R. We pass to Cartesian coordinates by setting X2lx\=x, xzlx\=y. After applying a transformation with coefficients in i?, corresponding to a" rotation of the axes, we may assume that the >'-axis is not parallel to any line joining two inflexion points of /=0. Then the abscissas a;i, . . . , x of the points of inflexion are distinct. By eUminating >^ and y* between the equations of the curve and its Hessian curve, we * We do not employ the fact, which now follows readily, that the coefficients of (6) are rational integral invariants of /. Xi y* 1 Xj yj 1 Xt yt 1 334 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIII obtain y expressed as a rational function <j){x) of x with coef- ficients in R. In fact, the ordinate of an inflexion point is uniquely determined by its abscissa. By substituting </)(x) for y in/=0, we obtain the equation X for the nine abscissas of the points of inflexion. For three collinear points of inflexion, = 0. Replacing yt by <t>{xi), etc., we obtain a rational relation (9) \l^{xi, xj, xi)=0, with coefl&cients in R. Conversely, if relation (9) holds for the abscissas of three points of inflexion, the latter are collinear. For, the line joining two of them, (xj, yj) and (xi, yt), meets the curve at a single point {x, y) and that point is a point of inflexion, so that \f/{x, Xj, a;t)=0 and X{x)=0 have a unique common solution x. Thus x = Xi and hence y = y<. Let a substitution of the group G of equation X for the domain R replace three roots Xi, Xj, Xt, for which (9) holds, by the roots Xr, Xs, Xt. By property B ( 149) of the group G, 4^(xr, Xs, Xt) = 0. Hence every substitution of G replaces the abscissas of three collinear points of inflexion by the abscissas of three collinear points of inflexion. Denote by 1, 2, 3 the points of inflexion on one side of a triangle of inflexion; by 4, 5, 6 those on a second side. Those on the third side may be denoted by 7, 8, 9 in such an order that 1, 4, 7 are collinear; 1, 5, 9 collinear, and hence 1, 6, 8 collinear. The third point on the line joining 2 and 4 is 8 or 9 (since 4 and 7 are collinear with 1) ; if it be 8, we interchange symbols 5 and 6, 8 and 9, and see that all the earlier collineari- ties are preserved, and that 2, 4, 9 are now collinear. Then the point on the line joining 2 and 6 must be 7 (since not 8 or 9), and that on the line joining 2 and 5 must be 8. We see that the 12 sets of collinear points of inflexion are those given by the 176] GROUP FOR THE INFLEXION POINTS 335 rows, columns, and positive and negative terms of the expansion of the determinant 1 4 7 2 5 8 3 6 9 Henceforth we shall denote the abscissas of the nine points of inflexion by the nine symbols [^77], where | = 0, 1, 2 and 77 = 0, 1, 2. Then the abscissas of collinear points of inflexion are those in the rows, columns, and positive and negative terms of [00] [01] [02] (10) [10] [11] [12] [20] [21] [22] and have the sum of their first indices divisible by 3, and also the sum of their second indices divisible by 3. Hence G is a subgroup of the group L of those substitutions on the nine roots which replace any three distinct roots [^jj], i = \, 2, 3, for which (11) 6+?2+f3=0, 771+172+173=0 (mod 3), by three distinct roots \^\ r}'i\ also satisfying congruences (11). We obtain a substitution of L if we take a b A B iO (mod 3), (12) ^'=ak-\-bv+c, v'=A^-\-Bv-\-C, where a, . . . , C are integers. For, 3 3 3 3 ^^'i = a^^i-^b'^Vi-\-^c=0, '^v'i^O (mod 3). <-i <-i i-i -i Further, the [|'i 17 'J are distinct. For, if {'i = {'2, i7'i=ij'2, then aiii-^2)-\-b(m-V2)^0, A{^i-^2)+B(r,i-v2)^0 (mod 3). But the determinant aB-bA is not congruent to zero. Hence ^1 = ^2, 171 =i?2, contrary to hypothesis. 336 INFLEXION POINTS OF A CUBIC CURVE [Ch.XVIU Conversely, every -substitution 5 of L is induced by a linear transformation (12) on the indices. Let 5 replace [00] by [c C] ; the same is true of the substitution P induced by (13) r^^+c, 77'^77+C (mod 3). Hence S = TP, where T is a substitution of L which leaves [00] unaltered. Let T replace [10] by [aA], where therefore a and A are not both zero. Thus we can find two integers b and B such that aB bA is not divisible by 3; if a^^O, we may take 5 = 1, J = 0; if -4?^0, we may take & = 1, 5 = 0. Then the substitution P' induced by (14) J' = a^+H ri'=A^-hBr,, aB-bA^O (mod 3) replaces [10] by [aA]. Hence T=T'P', where T' is a substi- tution of L which alters neither [00] nor [10] and hence not [20], in view of the first column of (10). Let T' replace [01] by [de], so that gj^O. The substitution Pi induced by leaves unaltered each [^0] and replaces [01] by [de\. Hence T'Pi~^ leaves unaltered each [^0] and [01] and is the identity. For, it leaves fixed [02] by the first row of (10), and hence [21] and [12] by positive terms of the expansion of determinant (10), and then [1 1] and [22] by the second and third rows. Hence T'=Pi and S=PiP'P, so that S is induced by a linear trans- formation (12). Now [cC] was any one of 3X3 roots, [a A] any one of 3^1 roots, and [de] any one of 3X2 roots. Theorem.* The group G oj the equation X Jor the abscissas {^7}] oj the nine points of inflexion is a subgroup of the group L of all of the 9X8X6 linear transformations (12) on ^, rj. The 3^ 1 incongruent linear homogeneous functions of ^ and 77 with integral coefl5cients modulo 3 are {, 17, i^+v), (-r?). * Jordan, TraiU des Substitutions, p. 302, where L is defined to be the group leaving (formally) unaltered the cubic function given by the sum of the products of the roots in each row, etc., of (10). But formal invariance may well intro- duce some confusion since the roots are not independent. For a wholly different determination of L, sec Weber's Algebra, 2d cd., vol. 2, p. 413. 177] GROUP FOR THE INFLEXION POINTS 337 Hence they are permuted by the linear homogeneous trans- formations (14), which form a group H. Since H permutes their four squares ^, . . . , (f t;)^, it is isomorphic with a group of substitutions on four letters. To show that the isomorphism is (2, 1), note that a transformation leaving ^ and rf unaltered at most changes the signs of | and tj. But if the sign of one is changed and the sign of the other is not changed, {^-{-nY is replaced by {k nY- Hence the identity / and J- k'=-^, v'^-v (mods) alone leave each of the four squares unaltered. The order of H is 48. There are 2X2X3 transformations (14) with b = 0, whence a^l or 2, B = l or 2, A=0, 1 or 2. There are 2X2X3^ transformations (14) with 6?^0, whence a and B are arbitrary, while aJ5 M = l then determines A. Hence * the quotient group H/{T, J} is simply isomorphic with the symmetric group on four letters. Thus H is a. solv- able group. The group T of the nine translations (13) is invariant under L. In fact, (14) transforms (13) into the translation ^' = ^-\-ac-\-bC, v'=v+Ac-\-BC. It follows that L is a solvable group. By 68, Ex. 4, or by 178, its subgroup G is solvable. Hence the equation X for the abscissas of the nine points of inflexion is solvable by radicals. 177. Group of the Resolvent Quartic Equation (6). Let fi, . . . , f4 be the roots of (6) and set >'i=rir2 4-^3^4, etc., as in Example 2, 150. Then yi, y2, ys are the roots of f - Qbf + 1 2b^y - 16a2 - 721^ = 0. Setting y = z-{-2b, we obtairi the reduced cubic z^=D, where D = lQ{a^+Al^)j^O, by (7). From f its discriminant, we see that p = (yi-y2)(yi-y.,i){y2-y:i)=DV-27. * Another proof follows from the fact that the linear fractional transformation on s= ^/fi, derived from (14) by division, permutes the values 0, 1, 2, oo of s. t Or from (s w2)(s w*z)(w2 *), where w^+v+lfO, 338 INFLEXION POINTS OF A CUBIC CURVE [Oh. XVIII Consider the special cubic form given by F in 175 with = 1, h= \. Then Z)= 48, so that P and each yi is irra- tional. Equation (6) is now (6') f4-6r2-f4r-3 = and is irreducible in the domain R{\) of rational numbers. For, no root is 1, 3, so that no root is rational. Further, a yi occurs in the coefficients of any quadratic factor, as shown by Ferrari's method of solving quartic equations. Hence the group of (6') for the domain R{\) is the symmetric group (Ex. 5, 153). Let/ be a cubic form whose ten coefficients are independent variables. Let R be the domain of the rational functions with rational coefficients of these ten variables. Then (Ex. 2, 153), the group of the quartic equation (6) for R is the symmetric group. 178. Group G of Equation X is the Linear Group L. After the adjunction of a root r of the resolvent quartic (6), the product of the equations of the three sides of an inflexion tri- angle has its coefficients in the domain {R, r), and the group G reduces to the subgroup which permutes the triples of abscissas of the points of inflexion on the sides of that triangle. First, let the triangle be that one whose sides contain the points of the rows in (10); these triples are merely permuted when the sign of either index is changed and also by the transformation and hence by the group of the 4-3^ transformations (12) with b=0, whose index under L is 4, By the interchange of the two indices, these triples are replaced by those in the columns of (10), so that the latter are merely permuted by the group of the transformations (12) with A=0. When b^A=0, we have ^' = a^+c, n^Brj+C, aB^O (mod 3). Unless a = B, the triples in the positive terms of the determinant (10) are replaced by those in the negative terms, since this is true for $' = ^, v''n, and since each transformation (15) ?' = ?+:, v'^v-\-C 178) GROUP FOR THE INFLEXION POINTS 339 permutes the three triples in the rows, the three in the columns, etc., of (10). Hence after the adjunction of the four roots of (6), the group G reduces to a subgroup 2 of the group of the 18 trans- formations (15). The group 2 is of order a multiple of 2. Take a = 2, h = Q in 175. Then (6) becomes r(r^ 8) =0, all of whose roots are in /?() =R{y/^Z). The Hessian ( 175) is fl'=-24a;2(x32+6xi2). After the adjunction of the four r's, the domain is J?(V 3), to which does not belong the irrationality * V^V 3 occurring in the sides of the inflexion triangle H = 0. The group 2 is of order a multiple of 3. For f=Xi^-\-2X2^-\-'iX3^-\-QXiX2X3, the Hessian is Q^h', where h'= o^i^ 2:^2^ 4a:3^ + 10a;iX2a;3. Then Zh' -\-rf has a Unear factor if r = 3, -1, -14V^. This is evident for r = 3. For r= 1, we get ^h'-f=4.{-Xi^-2X2^-^Z^ + QXiX2Xz), having the factor ^^X2 + ^XZ. V2 After the adjunction of the four r's, the domain is i?(V 3), to which "V^ does not belong. Thus the order f of 2 is a multiple of 3. We return to a cubic form / with arbitrary coefficients and the domain R defined by them. By the adjunction of the nine roots of X, the group G24 of the resolvent (6) is reduced to the identity. In fact, the abscissas and hence the ordinates of the nine inflexion points are in the enlarged domain. Thus the ratios of the coefficients of the equation of the line joining * No radical other than these two occurs in the sides of triangle (8) for r'^S. Hence the group G for this special cubic curve is of order 4. t Its order is in fact exactly 3; that of G is 6. 340 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIU three collinear inflexion points are in the enlarged domain; the same is true for their products by threes giving the inflexion triangles h-\-2irf, so that each root r of (6) is in that domain. It now follows from the theorem of Jordan ( 167) that the adjunction to R of the four roots of (6) reduces the group G of X for R to an invariant subgroup 2 of index 24 under G, such that G/2 is simply isomorphic with G24. This group 2 was shown to contain a transformation (15) of period 3, necessarily a translation (13). By interchanging ^ and rj if necessary, we may assume that ^ is. altered. Then the translation or its square is of the form Introducing ^ and 7; /^ as new variables, we obtain a group 2i conjugate with 2 under L and containing the translation ^' = 1+1, r}' = ri. The only transformations (12) which are commutative with this one are those with a = l, ^=0, Bf^O, b, c, C arbitrary, 2-3^ in number. The above translation is transformed into its inverse by ^'= ^, v'v- Hence exactly 4 3^ transformations of L transform into itself the cycUc group of order 3 generated by it. Since this number is one-fourth of the order of L, a subgroup of index 3 under L cannot trans- form this cyclic group into itself. But 2 is of order 6 or 18. In the first case, 2 contains a single cycUc group of order 3, which is therefore invariant under G; while G is of order 24-6 and hence of index 3 under L. Thus the first case is excluded by the preceding result. Hence 2 is of order 18 and G = L. Theorem.* If the coefficients of a cubic curve /=0 are independent variables, the group of the equation upon which depends the nine points of inflexion [^77], J, rj = 0, 1, 2, for the domain of the coefficients, is the group of all linear transformations on ^ and t? modulo 3. After the adjunction of the roots of the resolvent quartic (6), the group is that of the 18 transformations (15). The * Stated, but not completely proved, by Weber, Algebra, ed. 2, vol. 2. pp. 416-7. The proof is due to Dickson, Annals of Math., ser. 2, vol. 16 (1914), pp. 50-66. 1791 REAL POINTS OF INFLEXION 341 product P=A+24r/ of the linear functions which vanish at the sides of an inflexion triangle has as its coefficients quanti- ties in the enlarged domain. The determination of the linear factors requires the solution of a cubic equation. Consider the inflexion triangle associated with the rows in (10); after the adjunction of the roots of the corresponding cubic equation, the group permutes the roots [^17] in the same row. The only transformations (15) having this property are ^' = ^, ri'=rj-\-C, which form a group C3. The group of the resolvent cubic is therefore of order f -18 = 6. In the new domain, the group of the corresponding resolvent cubic for another inflexion triangle is C3. After the adjunction of one and hence all of its roots, we have the sides of two inflexion triangles, and their inter- sections give the nine inflexion points. Hence the determination of the inflexion points of an arbitrary cubic curve requires the extraction of a cube root and three square roots to solve the resolvent quartic equation, then the extraction of a square root and two cube roots to solve the two cubic equations which determine the sides of two inflexion triangles. IVo one of these three cube roots and four square roots can be avoided or ex- pressed rationally in terms of the others. 179. Real Points of Inflexion. Let the coefficients of the equation of the cubic curve be real. After a suitable choice of axes, the nine abscissas of the points of inflexion are the nine distinct roots of an equation with real coefficients ( 176). Hence at least one point of inflexion is real. The reduction to the form F in 175 can therefore be effected by a real trans- formation. By 177 the discriminant of the real quartic equation (6) is 21D^ and hence is negative. Thus * there are two distinct real and two imaginary roots. One of the real roots is positive and the other is negative, as shown by the values 00 , 0, -}- 00 of the variable r. By use of the same values we see that the slope of the curve y= a:* -1-66/^-1- . . . , corresponding to (6), is positive at the point whose abscissa is the positive root and negative at that with the negative root. At the points of inflexion (1, r, 5) the slope is -As^, by the * Dickson's Elem^nlary Theory of Equations, p. 45, or Ex. 5, p. 101. 342 INFLEXION POINTS OF A CUBIC CURVE [Ch. XVIII formula below (6). To obtain a real point, we must therefore take the negative real root r. Thus a real cubic curve F = has exactly three real points of inflexion, viz., (0, 0, 1) and (1, r, 5), where r is the single negative root. Any real cubic curve without a double point has exactly three real points of inflexion. CHAPTER XIX THE TWENTY-SEVEN STRAIGHT LINES ON A GENERAL CUBIC SURFACE AND THE TWENTY-EIGHT BITANGENTS TO A GENERAL QUARTIC CURVE 180. Existence of the 27 Lines. We shall first show that there is at least one real or imaginary straight Une (1) x = mz-{-n, y = pz-\-q on the general cubic surface (f){x, y, z)=0. Eliminating x and y, and equating to zero the coefficients of the resulting cubic function of z, we obtain four relations between the four param- eters m, n, p, q. These are consistent and have one or more sets of solutions, except possibly for special sets of coefficients of </). In fact, they are evidently consistent when <f>=xyz. See also 183. To determine the number of the straight lines on the cubic surface, we employ homogeneous coordinates, choosing the tetrahedron of reference so that 0:3 = 0, X4 = are the equa- tions of a line on the surface. Then no one of the terms xi^, 0:1^0:2, 0:10:2^, X2^ occurs in the equation of the surface, which is therefore of the form X3f-{-X4g = 0, where/ and g are homogeneous quadratic functions of Xi, . . . , X4. Part of the intersection of the surface by the plane X4 =CX3 is the line X3 = X4 = 0, and the remaining part is the conic /i+cgi=0 in that plane, where /i and gi are derived f rem / and g by replacing x^ by cxs. Hence in/i and gi, the coefficients of Xi^ are quadratic in c, the coefficients of XiXi and 0:2X3 are linear in c, and the coefficients of xi^, X1X2, xj^ are free of c. The conic degenerates into a pair of straight lines if and only 343 344 LINES ON A CUBIC SURFACE [Ch. XIX if the Hessian (discriminant) oi fi+cgi is zero. The degrees in c of the second partial derivatives of /i -\-cgi exceed by unity those of gi. Hence the degrees in c of the elements of the Hes- sian determinant are 1 1 2 1 1 2 2 2 3. Thus the determinant is of the fifth degree in c. There are cubic surfaces S for which this quintic has five distinct roots, so that the surface contains five pairs of straight lines intersect- ing the line C given by X3=Xi = 0. This is true if f = XiX2+eXiX4-{-tX2X^-\-aX3Xi-\-bX4^, g=Xi^-^X2XA. For then fi-]-cgi=cxi^-{-xiX2-\-cexiX3-\-{c^-\-tc)x2X3 + iac+bc^)x3^, whose discriminant is -2c[c*+2t(:^~{e-t^)c^+{b-te)c-\-a]. The second factor becomes any assigned quartic function by choice of t, e, b, a. Consequently, after the exclusion of the special surfaces for which the quintic is identically zero in c (for example, xyz = 0), or has fewer than five distinct finite roots, there remain surfaces of type S. For such a surface, each root c leads to a pair of lines A and B forming with C a triangle, which is the complete intersection of its plane X4 = cx3 with the cubic surface. The line C was any straight line on the surface. Hence every line on the surface is met by ten other straight lines lying on the surface. Any straight fine L on the surface meets the plane of the triangle ABC and meets it at a point on one of the sides, since the triangle is the complete intersection of its plane with the surface. Thus L is cither A, B, C, or one of the eight lines, other than B and C, which meet A , or one of the eight new lines meeting B, or one of the eight new lines meeting C. These 24 new lines are distinct, since otherwise one of them would meet 181] DOUBLE-SIX CONFIGURATION 345 two of the lines A, B, C, whereas the triangle ABC is the com- plete intersection of its plane with the surface. We readily exclude the case in which L passes through the intersection of A and B. For, if so, we may take those three lines as con- current edges of a tetrahedron of reference. Then xi = X2 = 0, :j^i=^3 = 0, X2 = :r3 = are lines on the surface, whose equation = therefore has no terms in X3 and X4 only, none in xo and X4 only, and none in xi and X4 only. Thus x^ occurs only in the terms Xia;2X4, XiX^Xi, 0:2^3^4. Hence the first partial deriva- tives of <^ with respect to each Xi vanish at (0, 0, 0, 1), which is therefore a singular point. But not every cubic surface has a singular point. Hence there are exactly 27 distinct straight lines on a general cubic surface. 181. Double-six Configuration. Consider a line h\ on the cubic surface and the five pairs a, c (? = 2, . . . , 6) of lines Fig. 16. on the surface which meet h\. The five planes hiatc^ are dis- tinct and three Unes on the surface do not concur ( 180). Hence no two of the lines C2, . . . , ce intersect. The locus of a line L intersecting C2, C3, C4 is a surface of the second order * * By choice of the axes, the equations of ci become y=mx, s=c, and those of Cj become y= mx, z=c. The line L joining the general point (a, ma, c) of C2 with the general point (6, mb, c) of cj is ab a+b xdz-\-ce, y = tnez+mcd, dm , e= . 2c 2c This meets Cz: x=h+t, y=rz-\-s, if idl)z+cet=0, (mer)z+mcds=0. In the determinant of the coefficients of z and the constants, replace d and by the values obtained by solving the equations of L. We get mxzcylmZ, cyzmc*xnUZ A= yzmcxrZ, mcxzc*ysZ '0, 346 LINES ON A CUBIC SURFACE (Ch. XIX and hence is an hyperboloid // of one sheet. By suitable choice of the axes, the equation of // is or tuvw = 0, if the binomials are designated /, etc. Hence \u = rw, { u = sv, \v = rt, \w = st are intersecting lines lying on H. When r varies, we obtain one set of generators; when 5 varies, we obtain a second set. Through each point of H passes one and only one generator of each set. We may assume that C2, cs, Ca belong to the first set of generators. Let Cs cut H at Pi and P2. Through P< passes one generator of the second set, which therefore meets C2, C3, C4, as well as C5. Hence C2, C3, d, C5 have two non-intersecting transversals, one of which is 61; call the second ai. Since ax contains four points of the surface (the intersections of a\ with C2, C3, C4, C5), it lies wholly on the surface. The line ai therefore meets one of the sides of triangle h\aQC&. As observed above, ai does not meet bi. Denote by cq that one of the remaining two sides which meets ai, and by ae the other side. Then no two of the lines ai, . . . , ao intersect. The line ai is met by five lines 62, . ^ . , fte of the surface besides C2, . . . , cq. No two of the lines h\, . . . , h& inter- sect. Now 62 meets one side of triangle hiazc^ other than C3, since cz meets the side a\ of triangle 0162^2- Hence 62 meets a^. The name double-six is given to this configuration of 12 lines Qt, bi {i = l, . . . , 6) such that no two a's meet, no two 6's meet, a< and bt do not meet, while at and bj meet if i^^j. 182. The 45 Triangles on the Cubic Surface. Since each line on the surface is a side of five triangles ( 180) and each triangle has three sides, there are 5-27/3 or 45 triangles. where Z=z*cK Suppressing the terms involving Z in A, we get c(mxz :y)* c{yz mcx)^= c{m^x^ y-)Z. Hence A is the product of Z by the required equation of degree 2. 183] GROUP FOR THE 27 LINES 347 Write cu=Cii for the c (j = 2, . . . , 6) in 181. Then aibiCii and OtbiCn are triangles on the surface. Let C23 be the third side of the triangle determined by a2 and 63. Hence C23 is distinct from each at and bt; also from Ci, since C2 is the only d meeting 02, and since C2 meets the side ai of aiftacs and hence does not meet 63. This new line C23 is thus not one of the ten lines meeting bi; nor does it meet cs, which intersects the side Z>3 of 0263^23. Hence C23 meets the side as of a3C3&i. Sim- ilarly, C23 meets &2- For like reasons, we may write Cu=Cfi for the third side of the triangle determined by a< and bj for ,y=i, . . . , 6; vy. We now have notations a<, bt, ctj for the 27 lines on the surface, and have the 30 triangles atbjCij. Next, if i, j, k are distinct, Ct] and <: do not meet. For, if they met, their plane would contain Oi and 6, whereas the latter do not intersect. Finally, two c's having four distinct subscripts intersect. For example, C34 meets one side of 0261^12, but does not meet 61 or a2, since 02 meets the side 63 of a\b2CzA\ hence C34 meets C12. Thus the sides of the remaining 15 triangles on the surface are c's with the following sets of subscripts: 12 34 56, 13 24 56, 14 23 56, 15 23 46, 16 23 45, 12 35 46, 13 25 46, 14 25 36, 15 24 36, 16 24 35, 12 36 45, 13 26 45, 14 26 35, 15 26 34, 16 25 34. 183. The Group for the Problem. Let R be the domain defined by the ratios of the coefficients of the equation for the cubic surface. After a suitable choice of axes of coordinates, we may assume that no two of the 27 lines (l) on the surface have the same parameter q. By eliminating m, n, p between the four relations mentioned in 180, we obtain an equation * E{q) =0 of degree 27 in q with coefficients in R. Since n may be eliminated last, it is a rational function of q with coefficients in R. The same is true of m and p. Thus the coefficients of the equation of a line on the surface are rational functions with *It may be assumed to have no multiple root ( 145 first foot-oote). 348 LINES ON A CUBIC SURFACE [Ch. XIX 1 m n 1 P Q 1 mi fix 1 Pi qi coefficients in R of the corresponding root q of E{q)=0. Hence we seek the group G oi E{q)=0 for R. Line (1) is in a plane with another line (2) x = miz-\-ni, y = piz-\-qi on the surface, if the eight parameters satisfy the relation obtained by eliminating x, y, z between the four equations: = 0. In view of the remark that m, n, p are rational functions of q, this yields a rational relation f{q, q\) = with coefficients in R. Let this relation be satisfied. Since there is a single line (5-2) on the surface coplanar with two intersecting lines {q) and (gi), fiq, 92) =0, /(^i, g2)=0 have a single solution 92. The next to the last step in the eb'mination of q2 gives ^2 expressed as a rational function of ^.and q\ with coefficients in R, say q2 = r{q, qi). Since any substitution of G must leave q2 r{q, qi) unaltered, it must replace the three g's corresponding to three Unes of a triangle on the surface by three q's of another triangle. Hence- forth we shall denote the 27 ^'s by the 27 letters a<, bj, Cy of 182. Thus G is a subgroup of the group F on the a<, bj, c^ which permutes the 45 triples having the same notation as the 45 triangles in 182. We readily determine the order and generators * of T. First, J = {aibi) . . . (aebe) interchanges the triples A^^aiVwa-nd A^, and leaves unchanged the 15 triples at the end of 182. Thus J is in T. * Much simpler and more natural than the generators used by Jordan, Traill desSubsliliitions, p. ^17. 183] GROUP FOR THE 27 LINES 349 Next, the 45 triples are evidently permuted by the substi- tution [ij] on the 27 letters which is induced by the interchange of the subscripts i and j. For example, [12] = {aia2){bib2)(Ci3C23){CuC24){Cl5C25){Cl6C26)' Finally, the group r contains A = {aia3C24)ia2CliCZ4){a'iCl2C23)(b3blC56)(bsC.i(iCi6){b6C35Cl5), which permutes the 45 triples as follows : (Ai2A32A42)(Ai4A34A24)(Ai3 A31 C13C24C56) (A43 A21 Ci4C23C56)(A23 A41 <: 12^34^56)^^, where P=VAi5 A36 Ci6C24C35)(A25 C14C25C36 C1QC25C34) (A51 A50 A53)(A45 C12C36C45 C16C23C45), while Q is derived from P by interchanging the subscripts 5 and 6. From these substitutions of T we evidently can derive one which replaces ai by any one of the 27 letters. If a substitu- tion of r does not alter ai, it must permute amongst themselves the pairs bj, cij{j = 2, . . . , 6) occurring in triples with ai. From B = (a5a3C24) (^2^45^34) (^4^25^23) (bsbscie) (61C36C56) (66C13C15), which is the transform of A by [15], and the [ij], we readily derive a substitution which leaves ai fixed and replaces 62 by any one of the ten letters bj, Cij {j>l). If a substitution of r leaves ai and 62 fixed, it leaves C12 fixed and replaces 63 by one of the eight letters bj, cij(J = Z, . . . ,6). Such a substitution can be derived from the [ij], i,j = Z, . . . , 6, and B. If a substitution of T leaves fixed ai, 62, C12, 63, it leaves fixed ci3 and permutes the pairs a, Ci2(i = S, . . . , 6) occur- ring in triples with ^2, and permutes the pairs at, Cts (' = 2, 4, 5, 6) occurring in triples with bs. Hence it permutes the letters C23, di, CLb, o,Q common to the two sets of pairs. Now [15][34] transforms A into (a6a4a23) (^2^35^34) (^3^25^24) (64&5C10) (61^40^50) (6o<^i4Ci8), 350 LINES ON A CUBIC SURFACE [Ch. XIX from which and the [y], ij = ^, 5, 6, we get a substitution leaving fixed ai, &2, Ci2, bs and Cis and replacing C23 by any one of the four letters C23, ^4, as, da- Next, the [ij], i,j = A, 5, 6, leave also C23 fixed and permute a4, a 5, a& in six ways. Finally, a substitution of V which leaves fixed oi, 04, as, ae, ^2, 63, C12, C13, C23 is the identity. For, by the above two sets of four pairs, it leaves fixed 03, Ci2, (I2, C13 (i = 4, 5, 6). Then by the triples containing C12, Ci3 (i = 4), it leaves fixed cse, c^e, c^s- By the triples containing one of the last three c's and C23, it leaves fixed Ci4, Ci5, C16. Hence it leaves fixed the fifteen c's and the six a's and consequently the six 6's, in view of the triples OibjCtj. The group T is generated by A, J and the [ij], i,j = l, . . . ,6 and is of order 27 10 8 24 = 51 ,840. It was noticed that J gives rise to an odd substitution on the 45 triples. Hence r has an invariant subgroup // of index 2 and order 25,920, composed of those substitutions of T which give rise to even substitutions on the triples. Various proofs * have been given of the simplicity of H. We shall next discuss the remarkable relation between the 27 lines on a general cubic surface and the 28 bitangents to a general quartic curve. From the group of the latter problem, we shall be able to conclude ( 189) that the group G of the former is identical with r. Thus G has the factors of com- position 2 and 25,920. In particular, the equation E{q) = for the 27 lines is not solvable by radicals. The equation has a resolvent of degree 45, corresponding to the 45 triangles, and a resolvent of degree 36, corresponding to the 36 double-sixes. But it has no resolvent of degree less than 27 other than the quadratic the adjunction of whose roots reduces the group to the subgroup // of index 2; this fact has been proved in three distinct ways.f The problem is thus of special complexity on the algebraic side. * Jordan, Trail4 des Substitutions, 444, 504; Dickson, Linear Groups, p. 307. t Jordan, TraitS des Substitutions, pp. 319-329; Dickson, Trans. Amer. Math. Soc, vol. 5 (1904), p. 126; cf. vol. 6 (1905), p. 48; Mitchell, ibid., vol. 15 (1914), p. 379. 184] CUBIC SURFACES AND QUARTIC CURVES 361 184. Relation between Cubic Surfaces and Plane Quartic Curves.* Using homogeneous coordinates, let f{x)=f{Xi, X2, Xs, X4)=0 be the equation of a cubic surface without a singular point, so that the first partial derivatives of/ with respect to xi, . . . , Xi are not all zero for a set of x's not all zero. The point iy-\-\z) on the line joining the points {y) = {yi, . . . , y4) and (2) is on the surface if f(y-\-\z)^f(y)+\L-\-h\^Q-[-\%z) =0, where, by Taylor's theorem, L = 2i^+ . . . +24^, Q = z,^^^+2z^Z2: ^^ ^ dyi dyi dyi^ dyidy2 Take (y) to be a fixed point P on the surface /=0. Then the line joining P and (2) meets the surface in two further points which coincide if (3) {lQy = Lf(z). Hence this equation of degree four in zi, . . . , 24 represents the tangent cone T4 to the surface /=0 with the vertex P on Any plane through a straight line / on the cubic surface meets the surface in I and a conic c. Let / and /' be the two intersections of / and c. The tangent to c at / is the Umit- ing position of a Une meeting the surface in three points and hence meets the surface at three coincident points. Thus / and this tangent line are principal tangent lines to the cubic surface and their plane is a tangent plane. Hence any plane through I is tangent to the surface at two points /, /', and thus is a bitangent plane. In particular, the plane through the fixed point P and any line / on the cubic surface is a bitangent plane to the surface and hence also to the tangent cone T4' The section of T^ by an arbitrary plane is a quartic curve C4. The intersections of E with the above 27 bitangent planes to Ti give 27 bitangent lines to C4. * Geiser, M athematische Annden, vol. 1 (1869), p. 129. 362 LINES ON A CUBIC SURFACE [Ch. XIX But there is an additional bitangent to d, making 28 in all. In fact, the plane L = passes through (y), in view of Euler's theorem (4), 173. Hence L = intersects T^, given by (3), in two pairs of coincident lines, the intersections of L = 0, Q = 0. Thus L = is a bitangent plane to T^. Before we can conclude that the general plane quartic curve has exactly 28 bitangents, we must show that, conversely, any given quartic curve C4 is the intersection of the plane U of the curve with the tangent cone to a suitably chosen cubic surface / at a point P on it. Let X, y, z be the homogeneous coordinates of a point in the plane V referred to a triangle of reference whose side s = is 0=0 Fig. 17. one of the bitangents to C4 and whose sides it; = and 3' = are any lines through the points of contact of this bitangent. Then the equation of C4 reduces to o^'f- = when z = and hence is (4) z<t>-x^y^ = 0, where </> is a cubic form in x, y, z. Let P be any point not in the plane U of C4. As the tetra- hedron of reference for homogeneous coordinates x, y, z, u of points in space, take that determined by the plane U and the planes through P and the sides of our triangle of reference. Then the desired cubic surface is (5) f=<t>+4uxy-\-AuH = 0. In fact, we shall proceed with this / as we did with the general / and find its tangent cone with the vertex P = (0, 0, 0, 1) in place of (y). Now dx dx '^' dy dy ^=9^+4u2, ^ = 4A:y+82. 92 82 du 184] CUBIC SURFACES AND QUARTIC CURVES 353 At the point P these derivatives vanish, with the exception of the third, which becomes 4. Hence L = Az. The second partial derivatives vanish at P, except Hence \Q = 2xy-\-^zu. Thus the tangent cone T^ from P is (2xy+42M)2 = 40/. Deleting the factor 4 and inserting the value of /, we get the equation (4) of C4 in the plane m = 0. EXERCISES 1. In the equation of the tangent plane to ^=0 at {U, . . . , /), the coeflScient of 2zi is 9yi* dyidyj 9yi9y4 which, for (/) = (y), equals '^^'a>'iVay./ 9yi Hence the tangent plane to ^ = at {y) is =0. 2. Let h, It, h be three lines on the cubic surface / which form a tri- angle and let P be a point of / not on one of the 27 lines. For i= 1, 2, 3, the plane Pit meets / in li and a conic ct. Call rj and 5 the points of intersection of U and ct; they are on T* and / and hence on ^=0. Thus the r<, Si are six points on the conic c in which Q=0 is met by the plane of /i, k, h- This plane meets the princii)al tangents /, ktof at P (the lines of intersection oi L=0, Q=0) at two points on c. Any plane cuts r in a quartic curve d. From A, /i, /j, we obtain bitangents /'i, I't, I' 3 to G whose points of contact are the projections of ft, Si (t=l, 2, 3) from P. From /i, /j, we obtain a bitangent t. The conic c projects into a conic c'. Hence C4 has the four bitangents I't, I'l, I't, T, whose eight points of contact are on a conic i'. 3. Thus T and any one of the 4') triangles on / determine a conic tf. But any one of the 28 bitangents can be used in place of r. Since any one of the conies is related in this way to each of the four bitangents, there 354 BITANGENTS TO A QUARTIC CURVE [Ch. XIX are 28-45/4 = 315 conies each containing the eight points of contact of four bitangents. 4. If (x, y, z) is a singular point with Z9^0 on (4), then (5) has the sin- gular point (x, y, z, u), where 2zu+xy=(). 185. Steiner Sets of Bitangents to a Quartic Curve. Let two sides, x = and >' = 0, of a triangle of reference for homo- geneous coordinates x, y, z be two bitangents to the quartic curve /=0. Then, for x = 0, f reduces to a perfect square, so that f=x<l>-\-{rz^-\-syz-\-ty^y, where <t> = c-^yq, c being a cubic form in x and z only, and q a quadratic form in x, y, z. Similarly, /v_o = {rz^-^uxz+vx^y. Hence the latter is identical with icc+r^z*. Thus f^xyq-\-{rz^-\-uxz+vx^yr^^+(rz^-\-syz-i-ty^y xyqi + {rz^+syz-\-ty^-\-uxz-\-vx^y, where qi =q 2{sz-{-ty){uz-\-vx). Hence /=0 can be given the form xyU=V^, where U and V are quadratic forms in x, y, z. Conversely, this equation evidently represents a quartic curve having x = and y = as bitangents. If X is any constant, the equation may be written as follows: xy{U-\-2\V-{-\^xy) = (F+Xa:^)^. Now U -\-2\V -\-\^xy is a quadratic form in x, y, z. Its discriminant (Hessian) is a determinant of the third order whose elements are linear in X with the exception of two ele- ments which involve X^. Hence the discriminant is of the fifth degree in X. It has no double root in general. For, if U = 2xz, V = ax^-\-bf-\-z^-\-2dxy-\-2exz, the discriminant is -|X[X*+8JX3+lG(J2-a6+5e2)x2-M66gX-{-4&], and the last factor can be identified with any quartic in X whose constant term is not zero. Hence if our quartic curve is general, 185] STEINER SETS OF BITANQENT8 356 there are five distinct values of X for which the quadratic form is a product of two linear functions. We thus have * Theorem 1. Given any two hitangents x = 0, y = to a gen- eral quartic curve, we can determine in exactly Jive ways a pair of lines ^ = 0, 77 = 0, such that the quartic becomes xy^rj = Q^. Then the eight points of contact of the four hitangents x, y, ^, t] lie on the conic Q = 0. Such a set of six pairs of hitangents is called a Steiner set (the older term was Steiner group). Let a,h] c, d; e,fhe three pairs of hitangents of the Steiner set determined by a, b. Then the quartic is abcd = Q^ or ab{cd-\-2\Q-^\^ab) = {Q-\-\aby. As above, we can determine X (Xp^O) so that cd+2\Q+\^ab=ef. Eliminating Q between this and abcd = Q^, we get ^abcd = ('\ab-{-j-^A\ Replacing a, c, e by a/X, \c, \e, we get (6) Aabcd = {ab-\-cd-efY, which may be written in the symmetrical forms (7) a'^y^+cH'^-\-^f^ = 2abcd-V2abef-{-2cdef, (7') v^+V^+V^=0. Transposing the first radical, we derive 4iCdef= {ef-\-cd aby. Hence the points of contact of c, d, e, f are on a conic. Theorem 2. The eight points of contact of any two pairs of bitangents of a Steiner set are on a conic. Thus tlie same Steiner set is determined by any one of its six pairs. Since the ^28 27 pairs of bitangents lie by sixes in a Steiner set, there are 63 Steiner sets. Also by 180 and Exs. 2, 3 of 184, with t and I'l as the given bitangents. 366 BITANGENTS TO A QUARTIC CURVE [Ch. XiX Three bitangents are said to form a syzygetic or an asyzygetic triple according as their six points of contact are or are not on a conic. Thus a, b, c are syzygetic if they belong to a Steiner set and two of them, as a and b, form a pair of that set. We shall prove the important Theorem 3. Three bitangents a, c, e of a Steiner set are asyzygetic if no two of the three form a pair. First, a, c, e do not all pass through the same point P. For, if they did, P would be a singular point of the curve (6). We may therefore take them as the sides of a triangle of reference and write b = la-\-inc-\-ne, d = pa-\-qc+re, f=sa-\-tc-\-ue. No one of the constants /, q, u is zero. For, if / = 0, for example, the lines b, c, e would meet at a singular point of the curve (6). Suppose that the points of contact of a, c, e with (6) lie on a conic C=ga^-\-hc^+ke^-{- ... =0. Then, by (6) for a = 0, etc., Ca~o = 'K\c{qc-\-re) e{tc-\-ue)], Cc~o = fJiW{la-\-ne) e(sa-\-tie)\, CemO = v{'^{la-\-fnc)c(pa+qc)]j where X, n, v are constants. Hence h = \q, g = til, g = vl, k = \u, k=fxu, h=vq, whence \= v, fi = v, \ = n, or \ = fi = v = 0. Thus g = h = k = and the conic C = passes through each vertex of our triangle of reference ace. The vertex a = c = is not on /=0, since u^O. But the points of contact of a with (6) are given by cd ef=0 and, being on C = 0, are the vertices lying on a; so that the vertex a = c = is on /=0. The assumption that Theorem 3 is false has thus led to a contradiction. Corollary. A Steiner set a, b; c, d; . . . has in common with the Steiner set a, c; b, d] ... no bitangent except a, b, c, d. 186] THE NOTATION OF HESSE AND CAYLEY 357 For, if e is common to them, a, c, c are asyzygetic by the first, and syzygetic by the second set. 186. The Notation of Hesse and Cayley. The 28 bitangents are designated [y] = [ii], i, j = \, . . . , 8; iji^j. We shall usually omit the brackets. We shall prove that the notation may be assigned to the bitangents in such a way that the 63 Steiner sets are given by (8) agah, bgbh, eg ch, dg dh, eg eh, fgfh; (9) abed, acbd, ad be, ef gh, egfh, ehfg; where a, . . . , h form a permutation of 1, . . . , 8. For examples, see 55 and Si below. Since g and h may be any two of the eight figures, there are 8-7/2 or 28 sets of the first type. Since there are 70 ways of selecting four figures out of eight, there are 35 sets of the second type, which is determined by a, b, e, d or by e,f,g, h. Having in mind also another appUcation (191), we shall proceed abstractly and consider the distribution of 28 symbols into sets of six pairs, in which the arrangement of the pairs and the order of the symbols of a pair are immaterial, and such that A. Every pair occurs in one and but one set. B. If a, /S, and 7, 5 are two pairs of a set, then a, 7, and /3, 5 are pairs of a set. C. Two sets a, /3; 7, 5; . . . and a, 7; /3, 5; . . . have only four symbols in common* For Steiner sets of bitangents, these properties hold in view of 185. As the first set we shall take Si. 12 34, 13 24, 14 23, 56 78, 57 68, 58 67. Rearranging the components of the first and fourth pairs by B, we see that there is a set S2 with 12 56, 34 78, and a set S3 with 12 78, 34 56, which, by C, have no further symbol in common with each other or with 5"!. Hence they introduce all of the 8+8 symbols not found in Si. It is at our choice to select the eight to go into S2; we take 52. 12 56, 15 26, 16 25, 34 78, 37 48, 38 47; 53. 12 78, 17 28, 18 27, 34 56, 35 46, 36 45. 358 BITANGENTS TO A QUARTIC CURVE [Ch. XIX The set Si with l2 57 and 34 68 has no further s3rmbol in common with Si and hence contains eight new symbols chosen from those of 5*2 and 53 not in the first or fourth pairs. The two of a pair cannot be two, as 26 and 25, from 52, since by B they would imply the pair 15 16, contrary to C. Again, if 25 occurs in 54, 16 does not occur; for, 25 jk and 16 /w would imply a set with 25 16 and jk Im, necessarily 52, whereas jk, Im are not in 52, by the last result. Hence after applying a substitution which permutes the eight symbols in 52 not in the first or fourth pairs, and the eight in 53 not in the first or fourth pairs, we may take 54. 12 57, 15 27, 17 25, 34 68, 36 48, 38 46. The set 55 having 12 25 contains, in view of 52 and 54, the pairs 56 16 and 57 17, and no further symbols from 52 and 54, and therefore six symbols chosen from 13 24, 14 23, 58 67, 18, 35, 45, 28, the earlier ones being paired for the sake of simplicity of refer- ence. But 57 17, 28/ imply a set with 17 28, 57/, contrary to 53. Hence 28 is excluded. If we take at least four of the first six symbols, we must pair two of them with each other; these two occur in 5i and cannot be a pair in Si, so that we have a contradiction with C. Hence we must take one and but one from each pair 13 24, etc. Since we may interchange the symbols of any one of these three pairs without altering sets 5i-54, we may assume that 55 contains 58, 13, 14. These and sets 5i-54 are not altered by (15 38) (26 47) (18 35) (27 46), (38 48) (47 37) (46 36) (35 45), which give rise to any permutation of 18, 35, 45. Hence we may take 55.- 21 25, 31 35, 41 45, 61 65, 71 75, 81 85. Without the use of further substitutions to fix the choice of equivalent notations, we can now prove that the 63 sets are 186) THE NOTATION OF HESSE AND CAYLEY 359 uniquely determined by properties A, B, C and that each is of one of the two types (8), (9). Using 52, ^4, Si, Ss, Si in turn, and property B, we get Se. 15 48, 26 37, 27 36, 18 45, 14 58, 23 67; 57. 15 38, 26 47, 27 46, 18 35, 13 58, 24 67; 58. 37 47, 38 48, 36 46, 35 45, 13 14, 23 24. The set 59 with 12 68 has, by 5i, 54, the pair 34 57 and the symbols 16, 26, 47, 37 paired with 28, 18, 35, 45, since not paired with each other in view of 52. But 16 is not paired with one of the last three in view of 55. Hence S9 contains the pair 16 28. A pair 26 35 would imply 47 18 by 5? and hence 37 45, contrary to Sg. A pair 26 45 would imply 37 18 by 56 and hence 47 35, contrary to Sr- Hence, by 56, 59. 12 68, 34 57, 16 28, 26 18, 37 45, 47 35. With 12 16 occurs 56 25 by 52, 55, and 68 28 by 59, while 67, 24, 23 are paired with 27, 46, 36, by 5i. But 67, 23 are paired with 27, 36 by 56, and 67, 24 with 27, 46 by 57. Hence 5io. 12 16, 56 25, 68 28, 67 27, 24 46, 23 36. By 52, 54, 53 we have the first four pairs of 5ii. 26 16, 15 25, 17 27, 28 18, 13 23, 24 14, and four of the s>Tnbols 13, 14, 23, 24, 58, 67. Pairs 58 /, 28 18 would imply 58 18, 28 1, contrary to 55. Pairs 67/, 17 27 would imply 67 27, tU, contrary to 5io. Finally, 13 14, 13 24 are excluded by Ss and Si. Hence we have Sn. By the same steps, 5i2. 47 16, 38 25, 17 46, 28 35, 58 23, 67 14; 5i3. 37 16, 48 25, 17 36, 28 45, 58 24, 67 13. By 57, 56, 5ii, 55 or 5io, Si, we get 5i4. 26 27, 47 46, 37 36, 16 17, 56 57, 78 68; 5i6. 15 18, 38 35, 48 45, 25 28, 56 68, 78 57. By 5i and 55, the set 5ifl with 12 58 has 34 67 and 25 18 and the symbols 15, 47, 37, 28, 46, 36, since 26, 38, 48, 27 are excluded by 59, Su, S13, Sio, respectively (since 25 18, 26 imply 360 BITANGENTS TO A QUARTIC CURVE [Ch. XIX that 25 is in a set Sq with 26 18). By 5i5, 15 is paired with 28 in Si6. By Sg and Su, 47 is not paired with 37 or 46. Hence 5i6. 12 58, 34 67, 25 18, 15 28, 47 36, 37 46. In exactly the same manner, we get 5i7. 12 13, 34 24, 25 35, 38 28, 26 36, 37 27; Si8. 12 14, 34 23, 25 45, 48 28, 26 46, 47 27. By .Si, 5io, Si6 we get the first three pairs of 5i9. 12 67, 34 58, 16 27, 17 26, 38 45, 48 35, and see that the further symbols are those in the last three pairs. By 5*4, 17 is not paired with 38 or 48, and by 53 not with 35 or 45. Hence 17 is paired with 26. By Ss and S15, 38 is not paired with 48 or 35 and hence is paired with 45. Similarly, using 5*1, 6*10, S12, Sn, we get 520. 12 24, 34 13, 16 46, 47 17, 15 45, 48 18, except as to the pairing of the last four symbols, which is deter- mined by So and S15. By Si, 5io, S13, Sis (and 5?, S15 for the pairing the final four symbols), we get 521. 12 23, 34 14, 16 36, 37 17, 18 38, 15 35. We may now determine uniquely the remaining 42 sets from 5i-52i by use of the simple property B. Thus by 52, 54, 521 and 5i6, 52o or 5i7, 5c, we get 522. 12 15, 56 26, 57 27, 58 28, 23 35, 24 45; 523. 12 38, 56 47, 57 46, 13 28, 23 18, 67 45. The remaining sets may be derived independently of each other from 5i, . . . , 523 by use of property B; they need not be tabulated here since they, like the earlier sets, are of one of the two types (8), (9). Theorem 4. There is a distribution of 28 symbols into 63 sets of 6 pairs such that the arrangement of the pairs in a set is immaterial and such that properties A, B, C hold. Any su^h distribution can be derived from that given by (8) and (9) by a substitution on the 28 symbols. 186) THE NOTATION OF HESSE AND CAYLEY 361 Hence the notation ij may be assigned to the bi tangents in such a way that the Steiner sets are given by (8) and (9). Since the sets 5i, 52, Si together contain all the bitangents, any bitangent syzygetic with 12 and 34 occurs in S\ (Theorem 3, 185). Similarly, every syzygetic triple is composed of three bitangents of a Steiner set, two of which form a pair. The converse was noted before Theorem 3. Any syzygetic triple of a Steiner set (8) is of the type ag, ah, bg; any one of a Steiner set (9) is of type ab, cd, ac or ab, cd, ef. The latter is repre- sented by 111, i.e., by three segments whose six end points are all distinct. The former is represented by the broken line n composed of the segments ba, ac, cd; and the same repre- sentation is given to ha, ag, gb. Theorem 5. A triple is syzygetic if and only if it is of type \\\ or type fl . Hence a triple is asyzygetic if and only if it is of one of tlie three types A, VI, \|/. Seven bitangents are said to form an Aronhold set if every triple contained in it is asyzygetic. An example is (10) 18, 28, 38, 48, 58, 68, 78, each triple of which is of type \\/ and hence asyzygetic. An Aronhold set is represented by a figure formed by seven segments with at most eight end points. It is not com- posed of more than two separate parts; for, if it be, it would contain three syzygetic bitangents |||. If a separate part has two or more points from each of which at least two segments radiate, that part is of type A, since otherwise it would contain a triple of type fl. Hence each separate part is a A or a fan Fn consisting of n segments radiating from one vertex. First, let there be a single part. Then the set is a fan F7. Its vertex may be any one of the points 1, . . . , 8, so that there are eight such Aronhold sets. That with the vertex 8 is given by (10). Finally, let there be two parts. If they are fans Fr and F the number of segments r-\-s is 7, and hence the number of points is f -M +5-|- 1=9, whereas at most 8 points occur. Hence one part is a A and the other a fan F4 ; for example, (11) 12, 23, 31, 48, 58, 68, 78. 362 BITANGENT8 TO A QUARTIC CURVE [Ch. XIX The triangle can be chosen in 56 ways (the number of ways of selecting three p)oints out of eight), and then the vertex of the fan is any one of the remaining five points. Hence (11) is one of 280 such sets. Theorem 6. There are exactly 288 Aronhold sets. Each is a Jan Fj or is composed oj a triangle and a Jan F\. 187. Group of an Equation for the 28 Bitangents. Take as the ic-axis of Cartesian coordinates a line not through the intersection of any two bitangents. Then the bitangents cross the :r-axis at 28 distinct points (^, 0), and each bitangent y = m{x ^ is uniquely determined by its f. If the equation to the quartic curve is F{x, y)=0, then F[x, m{x ^\ must be a perfect square in x. From the resulting two conditions on m and ^, we obtain w as a rational function of ^ and the coefficients of F{x, y), since there is a single m for each ^; and hence obtain an equation E{^) =0 of degree 28 whose coefficients are in the domain R of the rational functions of the ratios of the coefficients of F. Let G be the group of this equation for the domain R. Let ^ = 0, q = be the two bitangents determined by the roots ^1, ^2. Then (185) there are two conies U = 0, V = 0, with coefficients in the domain (R, ^i, ^2), such thsit F=pqU V^. As there proved, there are five values of X for which U+2\V-{-\^pq=0 is a pair of lines. Li the product P = n(C/+2XF+X2/>g), extended over these five values of X, the coefficients are in the domain (R, ^1, ^2). Now P is a product of ten linear functions of X and y, which represent ten bitangents forming with p and q a Steiner set. Settii\g y = in P = 0, we obtain an equation Ji^i, ^2, ^)=0, with coefficients in R, whose ten roots x are the ^'s of these ten bitangents. Thus/(^i, ^2, ^3) is zero when {1, h, h are roots of (^)=0 corresponding to a syzygetic triple of bitangents and not zero for three roots corresponding to an asyzygetic triple. In the first case, we shall call the roots syzygetic; in the second case, aszygetic. 187] GROUP FOR THE BITANGENTS 363 If a substitution 5 of G replaces three syzygetic roots {i, I2, ?3 by the roots ^'i, ^'2, ^'3, then, by property B of 149, /(^'i, $'2, ^'3)=0 and ^'1, ^'2, ^'3 are syzygetic. The converse is shown to be true by use of S~^ Theorem 7. Every substitution of G replaces syzygetic roots by syzygetic roots and replaces asyzygetic roots by asyzygetic roots. Corollary. Every substitution of G replaces any Steiner set by a Steiner set and every Aronhold set by an Aronhold set. We proceed to exhibit a fixed group r which contains G as a subgroup no matter how we vary the coefficients of the quar- tic equation and hence vary G. For the extreme case in which the coefficients are independent variables, we shall later see that G is identical with r. But in every case we shall in the meantime be able to determine the possible number of real bitangents from a knowledge of the substitutions of period 2 in r. The group G may contain a substitution S which interchanges the roots 18 and 28 of the Aronhold set (10) and leaves unaltered the remaining roots i8, i = 3, . . . , 7. If so, 5 replaces the Steiner set la 8a(a = 2, . . . , 7) by a Steiner set 2 in which the figure 8 occurs six times and hence is of type (8) with A = 8 (since we may permute g and h). Since 8 is paired in 2 with any figure except 2 and 8, we have g = 2. Hence 2 is 2a 8a (a = 1,3, . . . , 7), so that 5 replaces la by 2a if a > 2, and leaves 12 unaltered. Next, 5 replaces S by a Steiner set of type (8) with /j = 8 and containing 21 82, and hence with g = l, so that the new set is 21 82, la8a (a = 3, . . . , 7). Thus 5 replaces 2a by la if a>2. Finally, 5 replaces the Steiner set 31 81, 32 82, 3a 8a (a=4, . . . , 7) by one with 32 82, 31 81, and hence with 3a 8a, so that S leaves unaltered 3a (a = 4, . . . , 7). Similarly, S leaves unaltered ba {b, a = 3, . . . , 7; b9^a). Thus S is the product of the transpositions (ly, 2y),y = 3, . . . , 8, and is therefore the substitution on the 28 symbols [ij] which is induced by the transposition of the two indices 1 and 2. We obtain in this way 7! substitutions which leave unaltered the Aronhold set (10), merely permuting its seven symbols. 364 BITANGENTS TO A QUARTIC CURVE [Ch. XIX The group G may 'contain a substitution 5 which replaces the symbols (10) by the corresponding symbols of the Aron- hold set 17, 27, 37, 47, 57, 67, 87. If so, S replaces the Steiner set ia 8a (a = 1, . . . , 7; a^i), where i is a fixed integer = 6, by a Steiner set of type (8) with /f = 7, g = i, since 7 is paired with every integer except 7 and i. Hence the new set is ia7a (a = l, . . . , 6, 8; aj^i). Hence 5 leaves fixed ia (a ^ 6) and replaces i7 by i8. Thus S is induced by the transposition of the indices 7, 8. We now have 8! substitutions which permute the eight Aronhold sets typified by fans F7 with vertices 1, . . . , 8. These 8! substitutions on the 28 symbols are those induced by the 8! substitutions on the indices 1, . . . , 8, and form a group E. The group G may contain the substitution ^/18 28 38 48 58 68 78 ... \ ^^^^ \23 13 12 48 58 68 78 . . . /' which replaces the Aronhold set (10) by the Aronhold set (11) composed of the triangle with vertices 1, 2, 3 and a fan Fi with the vertex 8. Since P replaces the Steiner set la 8a (a = 2, . . . , 7) by one of type (8) with h = S and having the symbols 13, 12 and hence with g = l, P replaces 12 by 38, 13 by 28, and leaves unaltered la(a = 4, . . . , 7). Next P replaces the Steiner set 8a 4a (a = 1, 2, 3, 5, 6, 7) by that determined by 23 14 and hence leaves 24 and 34 unaltered and replaces 45 by 67, 46 by 57, 47 by 56. In this way we find that Pi238 = (12 38) (13 28) (23 18) (45 67) (46 57) (47 56), which may therefore be designated also by P45G7. Similarly, or by symmetry, we obtain the substitution i'aia,aia4 = -Pftft/SW5 = (l2 a3a4)(aia3 OC2a4,) (aia4 a2a3)(i3i/32 fi3^A)(fii^3 m^Wi^A ^2^3), where ai, . . . , /34 form a permutation of 1, . . . , 8. There are 35 such substitutions, since there are 70 combinations of 8 things 4 at a time. 188] NUMBER OF REAL BITANGENT8 366 We thus have a group, of order 288- 7! =36-8!, 188. Number of Real Bitangents. We shall employ the Lemma.* // an algebraic equation with distinct roots and with coefficients in a real domain R has exactly v pairs of con- jugate imaginary roots X2j-i, X2j(j = l, . . . , v), its group for R contains the substitution S={XiX2) . . . (x^j-iX^j) . . . (^2,-1X2,). If we apply the Corollary in 149 to the group |1, S\, we see that our Lemma is proved if we show that S leaves numeri- cally unaltered every rational function <l>{xi, . . . , Xn) oi the roots such that 4> has its coeflEicients in R and equals a quantity in R. Since the numerical value of <t> is real, remains numeri- cally unaltered when i = A/^ is changed into i in. each im- aginary root; but the resulting change in <j) is the same as if we had applied the substitution S. Corollary. Either the roots are all real or else the number of real roots equals the fiumber of letters mialtered by one of the substitutions of period 2 of the group for R of the equation. We shall prove that every substitution S of period 2 of the group r of 187 leaves fixed exactly 4, 8 or 16 of the 28 sym- bols. First, if 5 is in the subgroup E, it is induced by 1 , 2, 3 or 4 transpositions on the 8 figures and hence leaves unaltered f 16, 8, 4 or 4 symbols, respectively. Second, S = Pi234 leaves 16 symbols unaltered. Third, let 5 = a"^Pi234, where a is not the identity and is induced by the substitution which replaces 1, . . . , 8 by ai, . . . , ag. Then 5=5~^=-Pl234 <T = (rPaiatctuu ' Pl234=<rS = O^Pa^ata**- Hence (end of 187) 0^ is the identity and ai, a2, 03, a^ form a permutation of either 1, 2, 3, 4 or of 5, 6, 7, 8. In the latter case, <T is induced by (1 ai) (2 aa) (3 ag) (4 ^4), * A less precise theorem limited to irreducible equations was given by E. Maillet, Annates de Toulouse, ser. 2, vol. 6 (1904), p. 280. The present Lemma was given by Dickson, Annals of Matkemalics, ser. 2, vol. (190.')), p. 144. fThe verification can be made for the substitutions used below. 366 BITANQENT8 TO A QUARTIC CURVE [Ch. XIX SO that the substitution induced by (ai a2 as aA 5 6 7 8/ transforms 5 = aPi234 into that induced by (15) (26)(37)(48)Pi234, which leaves unaltered only the symbols 15, 26, 37, 48. In the former case, ai, . . . , a4 is a permutation of 1, 2, 3, 4. If this permutation is the identity, a permutes only 5, 6, 7, 8, and, after applying a transformation not altering P1234, we may take <r = (78) or (56) (78), whence S leaves 8 or 4 symbols unaltered. In the contrary case, we may assume that a is the product of (12) or (12) (34) by a substitution ci on 5, 6, 7, 8. If (Ti is the identity, we transform by (18) (27) (36) (45) and are led to the preceding case. There remain the cases <r = (12)(56), (12)(56)(78), (12) (34) (56) (78), for which 5 leaves unaltered 4, 4 or 8 symbols, and the case (12) (34) (56), which is equivalent to the second case. Theorem 8. There are exactly 4, 8, 16 or 28 real hitangents to a real guar tic curve without singular points* 189. Real Lines on a Cubic Surface. If we adjoin to the domain one root of the equation upon which depend the 28 bitangents to a quartic curve, the group reduces to a subgroup simply isomorphic f with the group of the equation upon which depend the 27 straight lines on the related cubic surface ( 184). The substitutions of period 2 of r which leave one symbol fixed leave unaltered 3, 7 or 15 of the remaining 27 symbols. Theorem 9. There are exactly 3, 7, 15 or 27 real straight lines on a general real cubic surf ace. X * It is then not of the tyf)e excluded in the proof of Theorem 1. The group discussion by Maillet {I.e., p. 323) is incomplete, as it fails to exclude the case of no real bitangents. Weber, Algebra, 2d ed., vol. 2 (1899), devotes pages 458-46.') to a proof that no other than these four cases can occur, and three pages to a proof that all four cases actually occur. For geometrical treatments see Zeuthen. Malhetnaticshe Annalen, vol. 7 (1874), p. 411; Salmon's Higfier Plane Curves, p. 220. t Note that the quotient of the order 288 -7! of T in 187 by 28 is the order of the group T in 183. t That these four, but no other, cases actually occur was shown by ScMdjii, Quarterly Journal of Mathematics, vol. 2 (1858), p. 117. 190) ACTUAL GROUP FOR THE BITANGENT8 367 190. Actual Group G for the 28 Bitangents. We have seen that G is a subgroup of the known group V and have applied this fact to the study of the reaUty of the bitangents. It is a proper conclusion of this investigation to give the proof that G is identical with r in case the coefficients of the quartic equa- tion are independent variables. For this purpose we need the following Lemmas, the proof of which differs from that given by Weber * mainly in a slight change pf notation made in the interests of symmetry, and in the elaboration of the proof of Lemma 2. Lemma 1. Given the seven bitangents of any Aronhold set of bitangents to a quartic curoe without singular points, we can determine rationally the remaining bitangents. Let (10) be the given Aronhold set and write Xi for [t8], and Xii for [y], i<8,y<8. We are to express the Xi] rationally in terms oi x\, . . . , xt. Since xia:23, 0:2X13, xzX\2 are pairs 18 23, 28 13, 38 12 of a Steiner set of type (9), we may mul- tiply X23, etc., by constants as in the derivation of (6), and obtain the equation /=0 of the curve in a form such that (12) f=^2XlzX2Xi2U^ = ^^X\2X\X2^1^ = 'iXiX23X2Xi3'U^, where (13) U=XiX23-\-X2Xi3-\-X3Xi2, ^^ =^1X23 -X2iCl3+a:3iCl2, w = rciX23 +X2X13 a^sici 2. Similarly, X2X12 and x^ia are pairs of a Steiner set (8) with g = S, h = l, and (12') f= ^2X12X4X14- q^f where ^ is a quadratic form. Then by (12i), ^2Xi2{x3Xi3-X4Xu) = ('U-q)iu-\-q). If one of the factors on the right were divisible by X2 and the other by X12, the intersection of X2 and X12 would be on a = -^ Algebra, ed. 2, vol. 2, 1899, p. 442. Down to this point in our eqxwiti<a of tiie thepry, we have made only an indirect use of Weber's treatment. 368 BITANGENTS TO A QUARTIC CURVE (Cu. XIX and hence be a singular point of /=0 by (12i). Hence, after changing the sign of q if necessary, we may assume that u q = 2\\X2X\2y where Xi is a constant not zero. Then u-\-q = 2{xzXi^-XaXi^/\i. Adding and replacing u by its expression (13), we get (14) XAX\A=X2X\z+\i^X2Xi2-\\{ XiX2i-\-X2Xiz-\-X-iXi2)' In the same manner (or by permuting 1, 2, 3 cyclically), we get , ,- [XAX2^=XiXi2-\-y^2^XzX2^-y<2{xiX2^X2XiZ-\-XzXi2), (14) \ \x^Xz^=X2X2Z-\->^7?X\Xi:i\z{XiX2Z-\-X2XizX^l2), where X2 and X3 are new constants not zero. To determine the X's, divide equations (14') by X2 and X3, respectively, and then add. We get the identity (15) ^y^+^\ =xY^'+X3:^i3-2:.23U:^23/, where / = X2X3+a:2/X3. Since xa, x\, 0:23 occur in the Steiner set (8) with g = 8, /f = 3, and no two are paired, they are not con- current (proof of Theorem 3) ; similarly, no three of the Xi con- cur. Hence /, x^, x\ are concurrent, so that x^ is a linear func- tion of / and x\. But (16) Xi = a\Xi+a2X2-\-azXz, where the a's are known constants each not zero. This sum must vanish for 1 = 0, a;i=0, whence a3 = a2X2X3. Thus X2 = Ai03, = h\a2, A3 where h\ is a new constant. Then, by (15) and (16), h,xJ'^+'^-h,X2^ =^1 N+^-//i(2+aiA0x23l. \ X2 X3 / L a2 a3 J 1901 ACTUAL GROUP FOR THE BITANGENT8 3C9 Hence, if i^i is a new constant, (.17; - \-- niX23 = T-xi, A2 A3 Hi a2 as Permuting 1, 2, 3 cyclically, we get k2X4 =12 ^.^3 _ ^2(2 + a2//2)xi3, as ai hx, = ^-\-'^-h3(2-taMxi2. ai a2 Since the three expressions for x^ must be identical, the three ^'s are equal and will be designated by k. Also, = 1 +aihi{2+aihi) = (1 +a<A,)2 {i = 1, 2, 3), so that ki= l/at. Thus (18) to=^+2l2+2i?. ai a2 as Since Xi is derived from Xs by permuting 1, 2, 3 cyclically, we have a2 as ai Al = , A2= , A3 = . as ai a2 Permuting 1, 2, 3 cycUcally in (17), we get (19') r^ \---=-kaiXi, A2 A3 fli X34 , Xi4 :J:13 , a;i4 , X24 X\2 , T \r=-. ka2X2, T" + T~ = "~; kasX3. A3 Al a2 Al A2 <l3 Adding and employing (18) and (16), we get -^-\-^+X^=-k{aiXi-\-a2X2+a^xz). Al A2 A3 Hence (19) ^J^-k[a2X2-k-a3Xs), ^ = ~-k{axXi+aiXi), Xi ai ' X2 a2 *34 X\2 . . T- = )i{a\X\-\-aixi). Xs ^3 370 BITANGENT8 TO A QUARTIC CURVE [Ch. XIX We may employ xs, Xq or xr in place of X4 in the preceding discussion. Instead of (16), we use (20) Xi = aiiXi-\-ai2X2+ 043x3 (i = 4, 5, 6, 7), where the o^ are known constants. Corresponding to (18) are (21) ki{aiiXi+at2X2-\-ai3X3) = -{-+ (j = 4, 5,6, 7), an 0x2 Ck3 in which ki, . . . , ki are constants to be determined. If the determinant J_ i_ i_| fli <2 a3 I were zero, where for example i = 4, 5, 6, then the corresponding Xi would be linearly dependent, whereas they were shown not to concur. Hence the ratios of four numbers h, . . . , I7 are de- termined by (22) k+lL+h+k^o 0- = i,2.3). ^ ^ a4j a5i aoj ajj \j y , / Multiplying the fth equation (21) by /< and summing, we see that the new right member is identically zero, so that 7 7 7 (23) ^kika,i = Q, 2^,/ifl<2 = 0, ^kUii = ^. <-4 f-4 i4 These equations determine the ratios of the ^'s, one of which is arbitrary and may be taken to be unity. Then (21) deter- mine X23, x\3, x\2, while (19) and the corresponding formulas determine xu, X2i, xst (j = 4, . . . , 7) rationally. We now have all the bitangents except 45, 46, 47, 56, 57, 67. But if we had used X5, xa, xi in place of x\, X2, x^, we would have ob- tained rationally all except 12, 13, 14, 23, 34, 34. The two steps together give all the bitangents a:^. Lemma 2. If we choose seven straight lines in a plane in a sufficiently general manner, we can determine rationally the equa- tion of a quartic curve without singular points for which the seven chosen lines form an Aronhold set of bitangents. 190) ACTUAL GROUP FOR THE BITANGENT8 371 For proof, we have only to reverse the argument made for Lemma 1, now taking the a^j as independent variables. Then 0:23, X13, a:i2 are determined rationally in terms of the o^ by means of (21)-(23), and then X4, X5, xe, xj, xu, xst, x^i, xjt are determined rationally by means of (20), (19) and the analo- gous equations mentioned above. Substituting the expressions for X23, ^13, X12 into (12i; and (13), we obtain the equation /=0 of a quartic curve, whose coefficients are rational functions of the Gfj. Its discriminant is not identically zero, since we saw in the proof of Lemma 1 that we can deduce equations (21)-(23) from the equation of a quartic curve without singular points. For i = 4, (20) and (21) give (16) and (18). From these and (19) we get (19'). Substitute the left member of the latter into (15). In the resulting term 0:4^23 Ai, replace X4 by its value (16); in the term kaiXiX4, Tep\a.ce kxA by its value (18). We obtain the right member of (15). Hence we have (15) and the equations derived from it by permuting 1, 2, 3 cyclically. From the relation between (14') and (15), it follows at once that (14) and (14') hold. Define q by the equation preceding (14). Then uq has the value indicated, so that (12') follows from the equation written below it. The equa- tions obtained from (12') by replacing 4 by 5, 6, 7 follow similarly from (20), (21) and the equations of type (19). Hence xa, X5, xg, X7 form with 'x\, X2, X3 an Aronhold set. Theorem 10. If the ratios of the 15 coefficients of a ternary quartic form f are independent variables and if R is the domain of the rational functions of these 14 ratios with rational coefficients, the group G for R of the equation E{^)=0 upon which depends the determination of the 28 bitangents to f=0 is the group V q/"187. It was proved in 187 that every substitution of G is in f. To prove the converse, it is sufficient, in view of the Corollar>' in 149, to show that every rational relation with coefficients in R between the roots of E{^)=0 is pre.served by each sub- stitution of r. To this end, let {1, . . , {7 be the roots corre- sponding to the bitangents 18, ... , 78. Since the latter 372 BITANGENTS TO A QUARTIC CURVE [Ch. XIX form an Aronhold set, the remaining 21 roots ^ = $^ {i,j=l, . . . , 7; iy^j) are rational functions of fi, . . . , {7 with coefficients in R (Lemma 1). If ci, . . . , cu are the ratios of the coefficients of /, any rational relation between the roots with coefficients in i? is of the type *Ul, . . , ?7, ?12, . . . , $67, Cl, . . . , Cl4)=0, where <l> is a rational function of its arguments with rational coefficients. First, we replace the f</ by their rational expres- sions in terms of the $*, Ct. Next, we replace each Ck by its rational expression (Lemma 2) in terms of the coefficients $, Tfii (i = l, . . . , 7) oi the seven bitangents of our Aronhold set. But these 14 quantities can be chosen at will. Hence after our replacements, relation $ = becomes an identity in the $, w. Thus <l> = remains true if we substitute for $1, . . . , $7 the seven roots in any order of any Aronhold set, provided of course we replace each $y by the root which arises from it by our substitution. But r is the group of all such sub- stitutions. Hence G = T. Corollary. The adjunction of one root of a certain equation of degree 36 reduces the group of the equation for the 28 bitangents to the group of the general equation of degree 8. In fact, the subgroup ( 187) of r is simply isomorphic with the symmetric group on 8 letters and is of index 36 under r. Since we know the generators of r and the representation of each substitution of r in terms of the generators (end of 187), we can prove by a straightforward argument that F is a simple group (Weber, I.e., pp. 454-6). 191. Symmetrical Notation for the Bitangents to a Quartic Curve. The separation of the Steiner sets into two types and likewise for the Aronhold sets was due to the lack of symmetry in the notation of Hesse and Cayley and not to a geometrical difference. A perfectly symmetrical notation was discovered * Riemann, Werke, 1876, p. 471. Weber, Theorie der Abelschen Functionen vom Geschlecht 3, Berlin, 1876, p. 82. Clebsch, " Ueber die Anwendung der Abelschen Functionen in der Geometrie," Crelle, vol. 63 (1864), p. 211, who used the notation (xi, xj, Xj; yi, yt, yi). Appell and Goursat, Thiorie des Fonclions Algibriques, 1895, p. 511. 191] SYMMETRICAL NOTATION FOR THE BITANGENT8 373 in connection with the theory of theta functions of odd character- istics (24) (xi^'i X2>'2 xzyz), where each rcj and >>< is or 1, and (25) x\y\-\-X2y2-\-xzy:i = \ (mod 2). 11x^ = 1 (mod 2), the congruence determines yz in terms of x\, y\, X2, y2, so that there are 2* such sets of solutions. If a;3=0 (mod 2), (26) xiyi+X2y2 = l (mod 2) has the four sets of solutions 0:2 = 1, y2 = lxiyi, and the two sets 0:2=0, y2=0 or 1, xi^yi = l; since >'3 = or 1, we obtain 2X6 sets of solutions of (25) with aca^O. Hence there are 28 symbols (24). Theorem 11. The 28 bitangents to a general quartic curve can he designated by the 28 symbols (24) in such a way that the 8 points of contact of the four bitangents A = {aibi 02^2 ^363), B = {cidi c^2 csds), C = (xiyi a;2>'2 xsyz), D = {ziwi Z2IV2 Z3W3) are on a conic if and only if (27) o,+ci+x,-f2=0, bt-\-di-\-yt+Wt^0{mod2) (t = l,2,3). This theorem, which leads to a symmetrical notation for the bitangents and presents the problem of the bitangents in a form suitable for extensive generalizations ( 192), was deduced in the papers last cited from the theory of abelian functions. We shall here give a very elementary proof,* depending upon two lemmas. Lemma 3. If A and B are any two distinct symbols (24), there exist exactly five pairs of symbols Co, Do ; . . . ; C4, D4, distinct from each other and from A and B, such that the sums of corresponding elements of the symbols A , B, Cj, Dj are all even, as in Theorem 1 1 . * Dickson, Bull. Amer. Math. Soc., vol. 20 (1914), pp. 403-4. 374 BITANGENTS TO A QUARTIC CURVE [Ch. XIX This lemma implies that the sums of corresponding elements of Co, Do, Cj, Dj, are all even. Thus the set of six pairs deter- mined by A and B is identical with the set of six pairs deter- mined by Co, Do. For such sets of six pairs, properties (A) and (B) of 186 therefore hold. The fact that also property (C) holds and hence also Theorem 11, may be stated as Lemma 4. The sets AB, CD, . . . and AC, BD, . . . have no further symbol in common. In the proof of these two lemmas, it suffices to consider symbols A, B having h^^ds (mod 2), and hence (after inter- changing A and B if necessary) with 63 = 0, d3 = l. For, if hz=dz, but hiT^di, the symbols A' = {azhz ^2^2 dibx) and B' , derived from A and B by interchanging the first and third pairs of elements, lead by the proof below to just five pairs C'j, D'j, from which we derive the required Cj, Dj, by interchanging the first and third pairs of elements. Next, if each bi=di, then ai^ci, for example, and we proceed as before with A* = {biai ^2^2 &3<i3), B* = {dici ^2^2 ^3^3). To prove Lemma 3, we may therefore assume that 63 = 0, d3 = l. If C and D are symbols for which congruences (27) hold, then y-i-\-W3 -{-1=0 (mod 2), so that either y3=0 or W3=0. Since the mutual order of C and D is immaterial, we may set 3^3 =0, whence 103 = 1 (mod 2). The conditions that C and D shall satisfy the condition (25) for a symbol are (26) and ziWi-\-Z2'W2-hz3 = l. By (27), the latter becomes 2 (28) ^{ai-\-Ct-\-Xt){bt-\-dti-yt)-\-a3+C3-\-X3 = l (mod 2), which determines X3 in terms of xi, yi, X2, y2. There are six sets of values of the latter which satisfy (26). One of these sets is Xi=ai, yi=hi {i \i 2), whence X3=a3, C=A (and hence D=B), since (29) C3 = l+Ciii+C2<f2 (mod2). Since this set is to be excluded, Lemma 3 is proved. For use in the proof of Lemma 4, we shall exhibit the five pairs Cj, Dj. In view of aibi-\-a2b2 = lx by and 62 are not both even. 192) FURTHER PROBLEMS OF CONTACTS OF CURVES 375 After interchanging the first and second pairs of elements in all of our symbols, if necessary, we may set 62 = 1. Then we may set A=(a h ah-\-\ 1 e 0), B = {cidi 7^2 C3I), in which, as well as below, cz is given by (29). The sets of solu- tions of (26), other than the above excluded set, are evidently the five sets of the first four elements in Co, ... , C4 below. After determining xz for each by use of (28), we see that the five pairs of symbols specified in Lemma 3 are C*=(l I kOzi, 0), Z>=(a+c,+l, 6+di+l, oA+l+c+A, dt+X, c+c+s*, 1) Cj=(a+1, 6, ai+6+1, 1, c+rf,+W,, 0) A=(ci+1, rfi, ct-\-b, d,, c,+rf,+W,, 1); Ci={a, b+l, ab+a+l, 1, e+Ci+adt, 0), Z),= (ci, di+l, Ct+a, dt. c+Ci+ad,, 1); C= (a+l, b+l, ab+a+b, 1, e+a, 0), Dt= {ci+1, di+1, Ci+a+b+l, dt, c,+a, 1) [a= Ci+di+d,+ l+adt+bd,]. To prove Lemma 4, we have to show that if C is one of these 10 symbols and E is one of the 8 not paired with C and not identical with C, the new set AC, BD, . . . does not con- tain E. If it did, there would be a symbol paired with E whose elements are the sums of corresponding elements of i4, C, E. But we readily verify that condition (25) is never satisfied for this symbol paired with E. After treating the cases in which C = Ci, we need not consider the cases C = Dt, since if P and Q form any pair of our ten symbols, A-{-B = P-\-Q = Ct+Dt imply A-\-Di-\-P=A+Ci-\-Q. Hence, treating C(ife = 0, 1) together, we need consider only six cases with = 2, four with C = C3, two with CC4, and C = Co, E=Di. For example, ^-f C2+G = (0 1 6-fi^ 5 0) is not a symbol satisfying (25). 192. Further Problems of Contacts of Curves. The pre- ceding symmetrical notation for the bitangents to a quartic curve is in accord with that used by Steiner * and Clebsch f in their treatment of a series of problems on contacts of curves. *JpurnaIfUr Math., vol. 49 (1855). t Ibid., vol. 63 (1864). 376 BITANGENTS TO A QUARTIC CURVE (Ch. XIX Let Cm be a real plane curve of order n having no double point. Set /> = i(n-l)(-2), Rj, = 2^''-^-2'-\ There are Rp curves of order w 3 having simple contact with C at (n 3)/2 points. The determination of these curves depends upon an algebraic equation E of degree Rp whose roots are designated by {xiyi xiji . . . Xpj^, where X\, . . . , yp form any set of integral solutions of x\y\-\-X2y2-\- . . . \-Xpyp=\ (mod 2). The simplest case is = 4; then /> = 3 and the problem is that of the i?3 = 28 bitangents to a quartic curve. For w^4, Clebsch proved that, if \i is any positive integer ^ Rp for which m( 3)/2 is an integer, the points of contact of Cn with the /i curves corresponding to the roots {x\y\ . . . x'py'^), . . . , (xi(''>>'i<''> . . . Xp^'^^yp^^) lie on a curve of order /i( 3)/2 if the congruences x\^-x'\^- . . . +a;p<^>=0, /p+3'"p+ . . . +>'p^^=0 (mod 2) (p = l, ...,/>) hold simultaneously. For = 4, the first case /i = 2 is evidently trivial, while the next case /x = 4 is the one treated in Theorem 11. The group * of equation E can be represented as a subgroup of the abelian f linear homogeneous group on 2p variables with integral coefficients taken modulo 2. Its substitutions of period 2 are conjugate to certain simple types, from which fact in connection with the Corollary in 188 we find that the number of real roots of equation E is one of the numbers 22p--(^ = l, ...,/>), 2=^-2^-i-2'-^ (y = 0, 1, . . . , tt), where Tr = p/2 or {p \)/2, according as p is even or odd. For = 4, we again get the number of real bitangents to a quartic curve ( 188). For = 5, we have /> = 6 and see that, of the 2016 conies tangent at 5 points to a quintic curve without * Dickson, Annals of Math., ser. 2, vol. 6 (1905), p. 146. t Leaving invariant a certain bilinear function of two sets of cogredient vari- ables. It is not a commutative group. 192] FURTHER PROBLEMS OF CONTACTS OF CURVES 377 double points, either all, 1024, 512, 480, 256, 128, 96, 64, 32 or none are real. These results, obtained so easily by group theory, are in complete agreement with those obtained by an elaborate geo- metrical proof by Klein.* The case in which the curve C has w(w 3)/2 double points was treated by Clebsch f and from the group standpoint by Jordan. f The latter treated also the group of the equation upon which depend the 16 singular tangent planes to Rum- mer's quartic surface with 16 singular points {I.e., pp. 313-5); also several technical problems of contacts of curves and the problem of the 16 straight lines on a quartic surface with a double conic (I.e., pp. 305-313). The various geometrical problems treated or mentioned in this Chapter and the preceding one have led to linear con- gruence groups. Such groups enter into the majority of the questions treated in Jordan's Traite des Substitutions and form the exclusive subject of Dickson's Linear Groups. * Math. Annalen, vol. 42 (1893), p. 3, p. 26; Rietnann'sche FUkhen, II (1892), pp. 117-255. t Journal fUr Math., vol. 64 (1865). X Traits des Substitutions, 1870, pp. 331-3. CHAPTER XX MONODROMIE GROUP 193. Definition of the Monodromie Group M. Consider an algebraic equation F{z, k)=Q in z whose coefficients are rational functions of the complex variable k. Let zi, . . . , Zn be the roots of F{z, ko) =0, where ^o is a constant. Let k vary- continuously from this initial value in any manner, but finally return to the same value ^o (ie., let the point representing k in the complex plane describe any closed path starting from and ending with the point representing ^o). Then the roots vary continuously and, after the circuit, take on their initial values in the same or a new order. Thus to each closed path corre- sponds a substitution on the roots. For example, if k describes a circle around the origin, the roots of s^ = 2k are interchanged. Two circuits may be combined into a single third circuit to which corresponds the product of the two substitutions corresponding to the two circuits. Hence the substitutions corresponding to all possible circuits form a group M, called the monodromie group * of F{z, k)=0 with respect to k. It was first studied by Hermite and Jordan. f 194. Monodromie Group an Invariant Subgroup of the Galois Group. Let <^ be a rational function of k and the roots zi, . . . , Zn, and let <}>, </>', <{>", ... be the functions derived from <f> by the various substitutions of If . If ^ = 0' = </>" = . . . ^ 4> is said to possess monodromie with respect to k. This is evidently the case with any <^ which equals a rational function * " Group in the function-theoretic sense," by Klein-Fricke. EUiptischen Modulfundionen, vol. 1, IS(K), p. 132; applications in vol, 2, 1892, p. 53, p. 599. t Traile des StibstUulions, 1870, pp. 277-9. 378 1941 MONODROMIE GROUP 379 of k, whether or not the coefficients of <t>(k, Zi, ..., 2) in- volve irrational constants. Let R be the domain defined by k and the coefficients of the powers of z in F(2, k). Any rational function <f> of the roots with coefficients in R, which equals a quantity in R, equals a rational function of k and hence is unaltered by every substitution of M. Then, by the Corollary in 149, Af is a subgroup of the Galois group G for R of the equation F = 0. Moreover, M is an invariant subgroup of G. For, let <t> be a rational function of the roots with coefficients in R which belongs to the subgroup M of index p under G. Then is a root of an equation E of degree v with coefficients in /?, so that <t> is an algebraic function of k. But <^ p>ossesses monodromie with respect to k. Hence <^ is a rational function f{k) of k with perhaps irrational coefficients. Replace the coefficients of f{k) by independent variables and substitute the result- ing expression in place of </> in E, and let the result be an identity in k. We obtain certain algebraic equations which the variable coefficients of f{k) must satisfy. Adjoin to R all of the roots of these numerical equations. Since <t> is in the en- larged domain, G reduces to a subgroup of M, necessarily M itself, since the group of monodromie is evidently unaltered by the adjunction of constants. But the adjunction of all of the roots of a second equation reduces the Galois group of the first equation to an invariant subgroup ( 167). A number of such adjunctions reduced G to Af ; whence M is invariant in G. For example, the Galois group for Rik) of 2* 2yt*=0 is the symmetric group Gt, since the equation is irreducible in R and the product of the dif- ferences of its roots is QVZk*. The only circuits causing a permutation of the roots are those aroimd the origin. Hence M is the cyclic group of order 3 and is invariant in G. 195. Applications of Monodromie. Jordan * employed monodrorriie to determine the Galois group of the equation for the n-section of the periods of elliptic f and hyj>erelliptic functions with 2/> periods. For the case of the trisection of Train des Substitutions, pp. 337-369. t Cf. K. Weber, Elliptische Functionen, 1891, p. 219. 380 MONODROMIE GROUP [Ch. XX the periods of hyperelliptic functions of four periods, the group is the quaternary abelian linear group modulo 3 and is iso- morphic * with the group for the equation of the 27 lines on a cubic surface (183). Monodromie has been applied to linear differential equations ^T.+/.(|S+ +/.(*) -0. For simplicity, the coefficients /(^) will be assumed to be rational functions of the complex variable k. Let 2i, . . . , 2 be a set of linearly independent solutions (integrals) and let ^o be a constant such that each 2 is an ordinary power series in ^ ^o- If now the point representing k describes a closed path starting from and ending with the point representing ^o, as in 193, the set of solutions zi, . . . , Zn becomes a set of solutions z'l, . . . , z', which are therefore linear functions of zi, . . . , Za with con- stant coefficients: z'l=ailZi+ . . . +ln3n, . , z'n=an\Z\-\- . . . +ann2n. With the chosen circuit is thus associated a linear transformation (75). The transformations obtained from all such circuits form a Unear group, called the monodromie group M of the differential equation. This group M is finite in case the integrals zi, . . . , z are algebraic functions of k. The theory of finite linear groups (Part II) is therefore applicable to the problem f of the deter- mination of all linear differential equations whose integrals are all algebraic. In the case just mentioned, M coincides with another important group G, which we proceed to define; but, in general, M is only a subgroup of G. According to Picard and Vessiot, the transformation group G of a linear differential equation with the linearly independent solutions zi, . . . , Zn possesses the following two characteristic properties (analogous to properties A and B of the Galois group of an algebraic equation, 149) : If a rational function F of Zi, . . . , z and their derivatives * Jordan, I.e., p. 369; Dickson, Linear Groups, pp. 306-7. t Jordan, Jour, fur Math., vol. 84 (1877), pp. 89-215. 1961 QUINTIC EQUATIONS, FORM PROBLEM 381 remains unaltered (as a function of k) by all the transformations of G, then F equals a rational function of k. Conversely, if such a function F equals a rational function of k, it remains unaltered by all the transformations of G. For the development and application of these concepts, see C. Jordan, Cours d' Analyse, vol. 3, 1896, pp. 193, 203, and L. Schlesinger, Handhuch der Linearen Differentialgleichungen, vol. 2, I, 1897, pp. 1-226, especially pp. 71, 96-102; vol. 2, II, 1898, pp. 148-159. 196. Quintic Equations, Form Problem. In the " form problem " for the icosahedral group, we are given the values of the fundamental invariants T, H,/, consistent with the relation between them ( 105, E), and require the values of the variables xi, X2. However, we desire primarily only their ratio z=xi/x2. Hence, given Z, we seek the values of z for which This icosahedral equation of degree 6o is remarkable both on account of the property that all its roots are linear fractional functions of a single root and the fact that the roots of any quintic equation can be expressed in terms of radicals and a root 2 of (1). In his Vorlesungen iiber das Ikosaeder, 1884, KJein therefore regards z as a new fundamental irrationality, a stage higher in algebraic complexity than radicals. More- over, z can be expressed in terms of elliptic modular functions {ibid., p. 132). Naturally there are many resolvent equations of degree < 60. For example, if we use the tetrahedral invariant t oi 105 and set r = i^/f, we get the resolvent of the fifth degree r(r2-l0r+45)2 + 1728(Z-i)=0. The form problem of the tenary linear group of order 168 ( 1 23, J, 125) is connected in a similar marmer with the equations of degree 7 whose Galois group is the simple group of order 168. For references to these and related subjects, consult Encyklo- padie der M athematischen Wissenschaften, vol. 1, I, pp. 533-552, 513-4; Weber, Algebra, ed. 2, vol. 2, 470-496, 530-550. SUBJECT INDEX (The numbers refer to pages) Abelian group defined, 62, 87 Abstract definitions of groups, 143 group defined, 52 Alternating group, 18, 43, 166 Anharmonic group, 65 Arithmetic substitutions, 12 Automorphisms of a group, 46 Average number of letters in all the substitutions of a transitive group, 32, of an intransitive group, 73 Axial group, 65 Base of an abelian group, 89 Bauer's theorem, 124 Bertrand's problem, 166 Cayley's table, 64 theorem, 64 Central of a group, 68 quotient group, 68 Characteristic subgroups and character- istic operators, 71, 109 Class of a substitution and of a sub- stitution group, 47 Cogredient isomorphisms, 68, 76 Commutative operators and commuta- tive groups, 54, 62 substitutions, 2, 18 Commutator quotient group, 69 Commutators and commutator sub- group, 68 Complementary groups, 67 Complete group, 46 set of conjugates, 22 Composite groups, 43 Conformal groups, 107 Congruence groups, 10 Congruent elements, 67 Conjoints, 37 Conjugate operators or subgroups, 59 substitutions and substitution groups, 5, 21 Construction of groups of order ^, 138 with invariant subgroups, 59 Contragredient isomorphisms, 76 Co-set and augmented co-set, 24, 66 Cross-cut of two groups, 23 Cyclic group, 7, 54 substitution, 16 Degree and order of a group, 2 of a substitution, 8, 16 Derived groupw, 69 DicycUc groups, 62, 170 Dihedral groups, 61, 168 Direct product, 13, 77 Divisible groups, 117 Divisor of a group, 3 Double co-set, 25 holomorph, 46 Doubly transitive groups, 40, 164 EUementary divisors of the order of an abelian group, 88 Elements of a group, 52, 68 Equivalent or congruent dements, 07 Even or positive subsUtutkms, 17 Examples of groups, 1 383 384 SUBJECT INDEX Factor groups, 176 Factors of composition, 174, 184 Four-group, 65 Frobenius's theorem, 77 Fundamental characteristic subgroup, 110 Generalizations of the groups of the regular polyhedrons, 152 Generating substitutions of a group, 7 Group defined, 2, 52 generated by two operators having a common square, 143 of a function, 5 isomorphisms, 46, 95, 101, 134, 160 the square, 4 property, 86 Groups involving no more than four letters, 41 only abelian subgroups, 112 of degree five, 45 movements of plane figures, 9 subtraction and division, 15, 81 the regular polyhedrons, 147 represented by matrices, 13 whose orders are px)wers of prime numbers, 118 divisible by 2 hut not by 4, 66 having simple abstract definitions, 143 Hamilton groups, 115 Holomorph of a group, 40 Icosahedron group, 150, 158 Icosian calculus, 159 Identity, 2, 53 Imprimitive substitution groups, 3S Independent cyclic subgroups, SS Index of a subgroup, 23 Indivisible group, 117 Inner isomori)hi3ms, 76, 183 Insolvable grou.js, 174, 180 Intransitive substitution group, 31 Intrinsic and relative properties of operators, 159 Invariant abelian subgroups of a prime- power group, 120 operators in the group of isomor- phisms of an abelian group, 101 subgroups, 21, 66 Invariants of an abelian group, 88 Inverse of a substitution, 11 an element or operator, 54 Isomorphisms defined, 33, 160 of the alternating and the sym- metric groups, 166 an abeUan group, 101 Jordan's theorem, 35 Kuhn's theorem, 37 Lagrange's theorem, 23 Left co-sets, 66 Maximal subgroup, 39 Metabelian group, 68 Metacyclic group, 12 Movements of the regular polygon, 9 Multiple isomorphisms, 34 Multiplication of substitutions, 2 Multiply transitive groups, 40 Nebengruppen, 24 Negative or odd substitutions, 17 Number of elements of a given order in an abelian group, 93 operators in a set of independent generators of a group of order />"* is invariant, 127 subgroups in a prime-power group, 123 of order p" in any group, 125 Octahedron group, 149, 154 Octic group, 4 Operator or operation of a group, 68 Order of a group and of an operator, 2,53 a substitution^ 8 the product of n operators, 70 Outer isomorphisms, 76 SUBJECT INDEX 385 Perfect group, 69 Period of an element of a proup, 53 Permutations and substitutions^ 36 0-subgroup, 71 P-isomorphisms, 134 Positive and negative substitutions, 16 Power of a group, 32 Primary groups, 1 18 Prime-power groups, 118 Primitive and imprimitive substitu- tion groups, 38 Product of two substitutions, 2 Quadratic group, 65 Quaternion group, 62 Quotient group, 34, 66 Rank of an abclian group, 92 Relative and intrinsic properties of operators, 159 Regular substitution group, 35 Representation of a group as a regular substitution group, 63 an abstract group as a transitive substitution group, 81 Right co-sets, 66 Roots of operators of an abelian group, 114 Self-conjugate subgroup, 21 Series of composition, 177 Set of independent generators of a group, 9, 90, 127 Similar substitutions and similar groups, 21 Simple group, 43 isomorphisms and simply isomorphic groups, 33, 73, 95 Simplicity of the alternating group, 43 Solvable group ,174 Subgroup, 3 Subgroups and quotient groups of an abelian group, 99 Substitution and substitution group, 1,2 groups of degree five, 45 Substitutions commutative with a given substitution, 19 Sylow's theorem, 27 Sylow subgroups, 27, 181 Symmetric group, 1, 3, 166 Systems of imprimitivity, 38 Tetrahedral group, 147, 152 Totient of a number, 11 Totitives of a number, 95 Transform of a substitution and of a substitution group, 20, 57 Transitive constituent of an intransi- tive group, 33 substitution group, 31 Transitivity of the symmetric group, 40 Transposition, 16 Vierergruppe, 65 PAKT n Abelian groups, canonical form of, 213 " Algebraic integer, 241 Canonical form, 196; theorems on, 212, 213 Change of variables, 203 Characteristic and characteristic equa- tion, 205 CollineatioDS and colUneation-groups, 198 Conjugate-imaginary groups, 209 Determinant of a linear transforma- tion, 194; theorems on the deter- minants of the transforroatiuns be- longing to a finite group, 196, 200, 202 Dihedral group, 220, 225 Diopbantine equation, 227 Exjuivalcnt groups, 202 Group-matrix, theorem oo, 268 386 SUBJECT INDEX Groups, binary, 215 , linear, 198 , linear fractional, 201 of coUineations, 198 linear transformations of deter- ninant unity, 200 the regular polyhedra, 220 order /<*, 231 order /-V, 272 degree n and class n 1, 274 , ternary, 235 Hermitian form, 207 invariant, 209 Hessian group, 239 Homology, 248 Icosahedral group, 223, 226 Identical transformation or the iden- tity, 196 Imprimitive groups, 228; theorem on, 229 Imprimitivity, sets of, 228 Intransitive groups, 206 Intransitivity, sets of, 206 Invariants (absolute, relative), 258; theorems on the number of, 258, 259 of the binary groups, 224 ternary groups, 253 Inverse of a linear transformation, 194, 210 Irreducible groups, 211 Linear transformation, 194 Matrix of a linear transformation, 194; theorems on the matrices of the transformations belonging to a finite transitive group, 271 Monomial groups, 229 Multipliers of a linear transforma- tion, 196; theorems on, 197, 257 Non-equivalent groups, 262; theorem on, 271 Octahedral group, 222, 225 Order of a linear transformation, 196 primitive group, 256 Power of a linear transformation, 196 Primitive groups, 228; order of, 256 Product of linear transformations, 195 Reducible groups, 210 Regular substitution groups, theorems on, 264, 265, 269, 272 Roots of unity, 239 Similarity-transformations, 196; the- orems on, 197, 202, 233, 234 Sylow groups, 231 Tetrahedral group, 222, 225 Transitive groups, 207; theorems on, 260, 261 Unitary form, 210 PAKT in Abel's theorem, 320 Abelian functions, 373 linear group, 376-7, 380 Adjunction, 299, 313-4, 317-20, 323-4, a38-41, 379 Aronhold set, 361, 367-71 Asyzygetic, 356, 361-3 Belongs to group, 296-7, 299 Binomial equation, 281, 312, 316 Bitangents, 350-76 Cardan's formulas, 301 Congruence group, 335, 377 Conies, 3.53-6, 373, 376 Conjugate, 297, 305 Constructions by ruler and compasses, 321-6 Contact, 353-77 Cross-ratio, 283 Cubic curve, 330-42 equation, 296, 301-3, 320-3 surface, 343-53, 366, 380 SUBJECT INDEX 387 Ciuve of order n, 376-7 Cyclic group, 306-9, 311, 316, 319,326 Cydotomic equation, 308-10 DifiFerential equation, 380 Discriminant, 290, 292, 302, 312 Domain, 280, 293, 299 Double point, 329 six, 345 Duplication of cube, 321-3 Elliptic function, 379, 381 Equality, 294 Equation with variable coefficients, 292 of the fifth and seventh degree, 381 Euler's theorem, 328 Factors of composition, 306, 311-2, 317,350 Fan, 361 Form problem, 381 Galois' criterion for solvability, 315 generalization of Lagrange's theorem, 298 theorem, 319 Galoisian equation, 320 resolvents, 284, 288 Gauss' lemma, 309 General equation, 292, 294, 317, 320, 372 Geometrical constructions, 321-6 questions, 321-77 Group of an algebraic equation, 286-9, 294, 300, 3a5, 333-41, 347, 362, 365, 367-71, 376-81 a differential equation, 380 on four letters, 290 Hessian, 329, 344, 354 curve, 330-3 Homogeneous coordinates, 327 Hyperboloid, 346 Hyperelliptic functions, 379 Icosahedral equation, 381 group, 381 Imaginary roots, 292, 365 Inflexion, 330-42 Integral root, 282 Invariant, 330, 333, 381 subgroup, 305-6, 318-0, 378 Irreducible, 280-1, 289, 292, 294, 297, 310, 320, 325 case for cubic equations, 320 Isomorphic, 304 Jordan's theorem, 317 Lagrange's formulas, 307 theorem, 298 Linear group, 335-8, 380-1 transformation, 328, 336, 380 Lines on a cubic surface, 343, 366 Monodromie, 378-81 Notation of Hesse and Cayley, 357 Primitive root, 308, 325 Quartic curve, 350-76 equation, 290, 296, 312-4, 337 surface, 377 Quintic equation, 381 curve, 376 Quotient group, 306, 318, 337 Radicals (see Solvable) Rational function, 284, 289, 294, 296, 298, 378, 380-1 root, 282 Reality, 341, 365-6, 376-7 Reciprocal equation, 290-2 Reducible, 280-1, 289, 291, 308 Regular group, 306, 309, 316 polygon, 321-6 Resolvent, 284, 290, 309-. 312. 350, 381 Root, definition of, 292 of unity, 307-11, 323-6 Series of composition, 306, 317 Simple group, 3(Nl, :i.y), 372. 381 Singular point, H29 388 SUBJECT INDEX Solvable equation, 301-3, 307, 310-4, 315, 317, 320, 337, 341 group, 279, 306, 310, 337 Steiner sets of bitangents, 354-77 Symmetric group, 295-6, 299, 305, 315, 317, 337-8, 364, 372 Symmetrical form for the equation to a quartic curve, 355 notation for bitangents, 372 Syzygetic, 356, 361-3 Tangent cone, 351-3 Theta functions, 373 Transitive, 289, 306 Triangle of reference, 328 on a cubic surface, 344-6 Trisection of an angle, 321-3, 326 Unaltered, 287, 289, 294, 296, 298, 38i Values of functions, 282, 314 AUTHOR INDEX (The numbers refer to pages) PAST I Abbati, P., 84 Abel, N H., 85 Alasia, C, 86 Battaglini, G., 86 Bertrand, J., 166 Betti, E., 15, 20 Bianchj, L., 86 BAcher, M., 13, 65, 86 Bolza, O., 33, 65 Burnside, W., 46, 65, 86, 123, 127 Capelli, A., 65, 160 Carmichael, R. D., 28 Cauchy, A. L., 12, 17, 23, 25, 27, 43, 47,85 Cayley. A., 64, 85. 159 Cebyggv, P. L., 166 Chapman, H. VV., 84 Cole, F. N., 86 Dedekind, R,, 69, 97, 115 Dickson, L. E., 117 Dyck, W., 81 Easton, B. S., 86 Elder, L., 84, 87 File, W. B., 68 Frattini, G., 35, 71 Frobenius, G., 25, 32, 71, 77, 87, 88, 113, 118 Galois, E., 2, 12, 24, 27, 45, 6ft, 85 Gauss, C. F., 11.8.5,87,89 Haffiilton, W. R., 45, 62, 115, 147, 150; 159 Heffter, L., 93 Hermite, C, 15 Hilton, H., 124 Holder, O., 34, 160, 176, 191 Huntington, E. V., 85 Jordan, C, 20, 34, 35, 50, 85,86, 160, 176 Klein, F., 15 Kronecker, L., 85, 89 Kuhn, H. W., 31 Lagrange, J. L., 4, 23, 84 Loewy, A., 183 Lucas, E., 98 Manmn^, W. A., 50 Mathieu, E., 106 Miller, G. A., 65. 114.117,138 Moore, E. H., 105, 160, 161 Moreno, H. C, 114 Netto, E., 86, 93 Pierpont, J., 65 Poincar6, H., 84 Ranum, A., 101 Remak, R., 176 Ruffini, P., 84 Schering, E., 89 S<5guier, J. de, 2 Scrrct, J. A., 166 Stickelbergcr, L.. 87, 88, 118 Sylow, L., 27, 119 Sylvester, J. J., 1 1 Vandermonde, A. J., 17, 84 Wcbcr, H., 12, 24, 34, 95, 08. 174 389 390 AUTHOR INDEX Bieberbach, L., 256 Blichfeldt,H.F., 231,256 Burnside, W., 231, 257, 268, 272 Dickson, L. E., 253 Frobenius, G., 256, 257, 274 Fuchs, L., 209, 215 Gordan, P., 215 Jordan, C, 207, 215, 239, 256 Klein, F., 215, 225, 254 Kronecker, L., 240 Loewy, A., 209 Maschice, H., 207, 253 Mitchell, H, H., 256 Molien, T., 257 Moore, E. H., 209, 250 Picard, E., 209 Schur, J., 256, 257 Valentiner, H., 209, 215, 256 Wiman, A., 254 Abel, N, H., 317, 320 Aronhold, S, 361, 367-71 Cardan, G., 301 Cayley, A., 357 Clebsch, A., 372, 375-7 Dickson, L. E., 320, 326, 350, 365, 373, 376-7, 380 Euler, L., 328 Galois, E., 298, 315-20, 378 Gauss, C. F., 309, 326 Geiser, C. F., 351 Hermite, C., 378 Hesse, O., 329, 357 Holder, O., 318 Jordan, C, 317, 336, 348, 350, 377-81 Kempner, A. J., 292 Klein, F., 377, 378, 381 Kronecker, L., 310 Lagrange, J. L., 298, 307 Maillet, E., 365-6 Mitchell, H. H., 350 Picard, E., 380 Riemann, B,, 372 Ruffini, P., 317 Schlafli, L., 366 Schlesinger, L., 381 Serret, J. A,, 320 Steiner, J., 354-77, 375 v Vessiot, E., 380 Weber, H., 320, 336, 366-7, 372, 379 381 Zeuthen, H. G., 366 QA Miller, George Abram 171 Theory and applications of M55 finite groups 1st ed. Pbywil ft Appiiitd ScL PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY 1^>,